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  • Course Introduction

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    • Time: 33 hours
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    Physics is the branch of science that explores the physical nature of matter and energy. Physicists examine the story behind our universe, which includes the study of mechanics, heat, light, radiation, sound, electricity, magnetism, and the structure of atoms. They study the events and interactions that occur among the elementary particles that comprise our material universe. In this course, we study the physics of motion from the ground up – learning the basic principles of physical laws and their application to the behavior of objects. Classical mechanics studies statics, kinematics (motion), dynamics (forces), energy, and momentum developed prior to 1900 from the physics of Galileo Galilei and Isaac Newton. We encourage you to supplement what you learn here with the Saylor course PHYS102 Introduction to Electromagnetism. Since mathematics is the language of physics, you should be familiar with high-school-level algebra, geometry, and trigonometry. We will develop the small amount of additional math and calculus you need to succeed during the course.


  • Unit 1: Introduction to Physics

    First, let's gain a basic understanding of the language and analytical techniques that are specific to physics. This unit presents a brief outline of physics, measurement units and scientific notation, significant figures, and measurement conversions.

    Completing this unit should take you approximately 2 hours.

    • Upon successful completion of this unit, you will be able to:

      • explain the difference between a theory and a law;
      • perform unit conversions using metric and traditional U.S. units;
      • compare and contrast accuracy and precision;
      • write numbers expressed in decimals as scientific notation and write numbers expressed in scientific notations as decimals; and
      • solve problems with the values of the most common metric prefixes.

    • 1.1: Scientific Theory, Law, and Models

      A scientific law briefly and succinctly describes an observed natural phenomenon or pattern. We often describe scientific laws as a single equation. For example, we describe one of Newton's Laws of Motion as F=ma. Because this is a brief, single equation, it is a law. Laws are supported by multiple, repeat experiments performed by different scientists over time.

      A scientific theory also describes an observed natural phenomenon or pattern, but in a less succinct manner. We cannot describe theories as a single simple equation. Rather, they explain the phenomenon or pattern. Charles Darwin's Theory of Evolution in an example of a scientific theory. The Theory of Evolution describes natural patterns, but cannot be described by a single equation. Like laws, theories must be verified by multiple, repeat experiments performed by different scientists.

      A scientific model is a representation of an object or phenomenon that is difficult or impossible to actually observe. Models provide a mental image to help us understand things we cannot see. An example of a model is the Bohr (planetary) model of the atom. This is a representation of an object (the atom) that is far too small for us to see. It allows us to develop a mental image so we can think about atomic structure.

    • 1.2: Physical Quantities and Units

      We define a physical quantity by how it is measured or by how it was calculated from measured values. It is either something that can be measured, or something that can be calculated from measured quantities. For example, the mass of an object in grams is a physical quantity because it is measured using a scale. The speed of a moving object in meters per second is also a physical quantity because it is based on two measured quantities (distance in meters, and time in seconds).

      The fundamental SI units are the kilogram (kg) for mass, the meter (m) for length, the second (s) for time, and the Ampere (A) for electric current. Derived SI units are based on the fundamental SI units. An example is speed, which is length per unit time.

      The metric system is a standardized system of units used in most scientific applications. The SI units are based on the metric system. The metric system is based on a series of prefixes that denote factors of ten. We call these factors of ten orders of magnitude. The prefixes tell us the relative magnitude of the measurement with respect to the base unit. Because the metric system is based on these powers of ten, it is a convenient system for describing measurements in science.

      • book icon
        Physical Quantities and Units Book
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        Read this text to review these physical quantities and units. Make sure you are familiar with the prefixes, symbols, values, and examples in Table 1.2. We will use them frequently during this course.

    • 1.3: Converting S.I. and Customary U.S. Units

      If we want to know how many cm are in 5.0 m, we can use dimensional analysis to convert between meters and centimeters. To do this, we use the prefix's order of magnitude as a unit conversion factor. Unit conversion factors are fractions showing two units that are equal to each other. So, for our conversion from 5.0 m to cm, we can use a conversion factor saying 1 cm = 10-2 m (again see Table 1.2 from section 1.2). To determine how to write this equivalence as a fraction, we need to determine what the numerator and the denominator should be. That is, we could write 1 cm / 10-2 m, or we could write 10-2 m / 1 cm.

      We can determine the proper way to write the fraction based on the given information. When performing dimensional analysis, always begin with what you were given. Then, write the unit conversion factor as a fraction with the unit you want to end up in the numerator and the unit you were given in the denominator. This will result in the answer being in the unit you want.

      {\mathrm{what\ you\ were\ given}\times\frac{\mathrm{unit\ you\ want}}{\mathrm{unit\ you\ were\ given}=\mathrm{unit\ you\ want}}}

      For our example, we want to determine how many cm are in 5.0 m. The given is 5.0 m. The unit we want is cm, and the unit we were given was m. So, we would set up the conversion as {5.0\ \mathrm{m}\times\frac{1\ \mathrm{cm}}{10^{-2}\ \mathrm{m}}=500\ \mathrm{cm}} . The meter unit cancels out in this calculation. Because the meter unit cancels out, we are left with cm as the unit of the answer.

      The same use of dimensional analysis also applies to non-metric units used in the United States. For example, we know that one foot equals 12 inches. These length measurements are not part of the metric system. We can determine how many inches are in 5.5 feet using the same dimensional analysis technique, where {5.5\ \mathrm{feet}\times\frac{12\ \mathrm{inches}}{1\ \mathrm{foot}}=66\ \mathrm{inches}} .
      • book icon
        Unit Conversion and Dimensional Analysis Book
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        Read this text for examples of how to calculate physical quantities and units of measurement.

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        Metric Units, Converting Units, Significant Figures Page
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        This lecture accompanies what you just read. At the end of the video, Greg Clements discusses significant figures which we cover in the next section.

    • 1.4: Uncertainty, Accuracy, Precision, and Significant Figures

      Uncertainty exists in any measured quantity because measurements are always performed by a person or instrument. For example, if you are using a ruler to measure length, it is necessary to interpolate between gradations given on the ruler. This gives the uncertain digit in the measured length. While there may not be much deviation, what you estimate to be the last digit may not be the same as someone else's estimation. We need to account for this uncertainty when we report measured values.

      When measurements are repeated, we can gauge their accuracy and precision. Accuracy tells us how close a measurement is to a known value. Precision tells us how close repeat measurements are to each other. Imagine accuracy as hitting the bullseye on a dartboard every time, while precision corresponds to hitting the "triple 20" consistently. Another example is to consider an analytical balance with a calibration error so that it reads 0.24 grams too high. Although measuring identical mass readings of a single sample would mean excellent precision, the accuracy of the measurement would be poor.

      To account for the uncertainty inherent in any measured quantity, we report measured quantities using significant figures or sig figs, which are the number of digits in a measurement you report based on how certain you are of your measurement. Reporting sig figs properly is important, and we need to account for sig figs when performing mathematical calculations using measured quantities. There are rules for determining the number of sig figs in a given measured quantity. There are also rules for carrying sig figs through mathematical calculations.

    • 1.5: Scientific Notation

      Often in science, we deal with measurements that are very large or very small. When writing these numbers or doing calculations with these physical quantities, you would have to write a large number of zeros either at the end of a large value or at the beginning of a very small value. Scientific notation allows us to write these large or small numbers without writing all the "placeholder" zeros. We write the non-zero part of the value as a decimal, followed by an exponent showing the order of magnitude, or number of zeros before or after the number.

      For example, consider the measurement: 125000 m. To write this measurement in scientific notation, we first take the non-zero part of the number, and write it as a decimal. The decimal part of the number above would become 1.25. Then, we need to show the order of magnitude of the number. We count the number of decimal places from where we placed the decimal to the end of the number. In this case, there are five places between the decimal we put in and the end of the number. We write this as an exponent: 10^{5} . To put the entire scientific notation together, we write: 1.25 \times 10^{5}\ \mathrm{m} .

      We can also do an example where the measurement is very small. For example, consider the measurement: 0.0000085 s. Here, we again begin by making the non-zero part of the number into a decimal. We would write: 8.5. Next, we need to show the order of magnitude of the number. For a small number (less than one), we count the number of places from where we wrote the decimal back to the original decimal place. Then, we write our exponent as a negative number to show that the number is less than one. For this example, the exponent is: 10^{-6} . To put the entire scientific notation together, we write: 8.5 \times 10^{-6}\ \mathrm{m} .

      We can also convert values written in scientific notation to decimal notation. Consider the number: 5.0 \times 10^{3}\ \mathrm{m} . We can write this as normal notation by adding the appropriate number of decimal places to the number, past the decimal written in scientific notation. Here, the order of magnitude (number of decimal places) is three, as we see from the exponent part of the number. Because the exponent is positive, we add the decimal places to the right of the number to make it a large number. The value in normal notation is: 5000\ \mathrm{m} .

      We can also do this for small numbers written in scientific notation. Consider the example: 4.2 \times 10^{-4}\ \mathrm{m} . We can write this as normal notation by adding the appropriate number of decimal places to the left of the number to make it a small number. Here, we need to have four decimal places to the left of the decimal in the scientific notation. The value in normal notation is: 0.00042\ \mathrm{m} .
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        Review of Scientific Notation Page
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        This video describes how to convert between standard and scientific notation. Pay special attention to how Jennifer determines the exponent when writing out the scientific notation.

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        Applying Scientific Notation Page
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        This video shows some practical applications to scientific notation. Pay close attention to the various ways of writing the same number.

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        Converting Scientific Notation to Standard Notation Page
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        Watch this video to review converting scientific notation to standard notation.

    • Unit 1 Assessment

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        Unit 1 Assessment Quiz
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        Take this assessment to see how well you understood this unit.

        • This assessment does not count towards your grade. It is just for practice!
        • You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
        • You can take this assessment as many times as you want, whenever you want.

  • Unit 2: Kinematics in a Straight Line

    We begin our formal study of physics with an examination of kinematics, the branch of mechanics that studies motion. The word "kinematics" comes from a Greek term that means "motion". Note that kinematics is not concerned with what causes the object to move or to change course. We will look at these considerations later in the course. In this unit, we examine the simplest type of motion, which is motion along a straight line or in one dimension.

    Completing this unit should take you approximately 5 hours.

    • Upon successful completion of this unit, you will be able to:

      • compare and contrast distance and displacement;
      • define and distinguish between vector and scalar physical quantities;
      • explain the relationship between instantaneous and average values for physical quantities;
      • compare and contrast speed and velocity;
      • solve one-dimensional kinematics problems;
      • describe the effects of gravity on an object in motion;
      • calculate the position and velocity of an object in free fall; and
      • draw and interpret graphs for displacement and velocity as functions of time, and determine velocity and acceleration from them.

    • 2.1: Vectors, Scalars, and Coordinate Systems

      A scalar physical quantity is a measurement of quantity that has a magnitude (amount), but not a direction. Examples of scalar quantities include mass and temperature; no direction is associated with these measurements. Distance is also a scalar quantity because it has no direction associated with it.

      A vector physical quantity is a measurement that has a magnitude (amount) and direction. Vectors are often depicted as an arrow. The length of the arrow shows the magnitude of the quantity, and the direction of the arrow shows the direction of the vector.

      For simple one-dimensional systems, a vector is often written as the magnitude with a (+) or (−) to indicate direction, with (+) going toward the right and (−) going toward the left. Displacement and velocity are examples of vector quantities. For example, 5.5 km/s east. This measurement shows the magnitude of the velocity (5.5 km/s), and the direction (east).

    • 2.2: Instantaneous and Average Values for Physical Quantities

      An instantaneous value is a value measured at a given instant, or time. For example, we can measure the velocity of an object right at high noon as 5.5 km/s east. This is an instantaneous value because we measured it at a given instant in time. A car's speedometer is an example of an instantaneous measurement. Furthermore, velocity does not necessarily stay constant over time, so instantaneous measurements can vary depending on when you take the measurement. 

      An average value is calculated over a period of time. For example, to calculate average speed, divide the distance traveled by time traveled. For example, if you drive 30 miles in two hours, your average speed is 15 miles/hour. However, as we know from driving, we rarely drive exactly the same speed for two hours. So, the instantaneous value of your speed could vary at any given time, but the average value is still 15 miles/hour.

      • book icon
        Time, Velocity, and Speed Book
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        Read this text which explains what we mean when we talk about instantaneous versus average time, velocity, and speed. Note that we will explore the calculations you see in this text in detail in Section 2.4 below. For now, make sure you understand what these concepts mean before we begin discussing distance and displacement next.

    • 2.3: Distance and Displacement

      Distance describes how much an object has moved. It depends on how the object has moved, that is, the path the object took to get from the starting point to the ending point. The units for distance pertain to length, such as meters. Distance is a scalar quantity because it describes the magnitude of the measurement, but not a specific direction.

      Displacement describes an object's overall change in position. It only depends on the starting and ending points of the object. It does not depend on the path taken to get between the two points. Like distance, the units for displacement are also length, such as meters. However, displacement is a vector quantity, which means it has a magnitude and a specific direction associated with the measurement. So, the complete value for displacement must also include a direction.

      For an example, consider a four-story building. A person needs to travel on the elevator from the first to the third floor. To accomplish this, the person could take an elevator directly from the first floor to the third floor. In this case, the distance and displacement are the same, because the person went directly from the starting to the ending point.

      However, this is not the only way the person could travel from the first to the third floor. They could accidentally hit the fourth floor button when they got on the elevator. In this case, they would travel from the first floor to the fourth floor, and back down to the third floor. In this instance, the displacement is still from the first floor to the third floor. But, the distance is longer, because the person took a detour to the fourth floor before going back down to the third floor.

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        Displacement Page
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        Read this text, which discusses position and the difference between distance and displacement.

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        Distance and Displacement as Scalar Vectors Page
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        This lecture accompanies what you just read.

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        Displacement vs. Distance Page
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        This video demonstrates the difference between distance and displacement, and their associated graphs with time.

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        Displacement Time Graphs Page
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        Watch this video to learn how to differentiate distance and displacement, and graph them with respect to time.

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        Vectors and Scalars Page
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        Watch this video to review scalars, vectors, and how they connect with distance and displacement.

    • 2.4: Speed and Velocity

      Elapsed time, \Delta t, is the change in time. Elapsed time is calculated as \Delta t=t_{f}-t_{i} , where t_{f} is final time and t_{i} is initial time. The Greek letter delta, Δ, means change. So, Δt means change in time. You will see this frequently in this course. When calculating elapsed time, we often assume the initial time is zero, to make the subtraction easier.

      Average velocity is the displacement divided by the elapsed time: \vec{v}=\frac{\Delta x}{\Delta t}=\frac{x_{f}-x_{i}}{t_{f}-t_{i}} . Here, the line above the v shows that it is an average quantity. This is the common notation for average quantities. To calculate the average velocity, divide the change in displacement by the elapsed time.

      The average velocity is a vector quantity because displacement is a vector quantity. Because we calculate average velocity from a vector quantity, it itself is a vector quantity. This means that average velocity has a direction associated with it. In one-dimensional systems, this means that the average velocity is written with a (+) or (−) sign, depending on the direction of the displacement.

      Instantaneous speed is the magnitude of the instantaneous velocity, measured at a given time or instant. Unlike velocity, instantaneous speed is a scalar quantity, so it does not have a direction associated with it. For example, if the instantaneous velocity of an object is −2 m/s in one-dimensional motion, the object's instantaneous speed is simply 2 m/s.

      The average speed of an object is the object's distance divided by the elapsed time. This is similar to the average velocity, which is the object's displacement divided by the elapsed time. Recall that distance is a scalar quantity that describes how much an object moved and that it can be very different from the vector displacement. Therefore, the average speed of an object is also a scalar quantity, and it can differ from the average velocity.
      • book icon
        Time, Velocity, and Speed Book
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        Now, let's return to this text which you read in Section 2.2, to study the calculations in more detail. Pay attention to Figure 2.10 and Figure 2.11 which show examples of how to calculate displacement, distance, average speed, and velocity.

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        More on Speed and Velocity Page
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        This lecture accompanies what you just read.

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        Speed and Velocity vs. Distance and Displacement Page
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        Watch this video to discover how average speed and velocity relate to distance and displacement.

    • 2.5: Motion with Constant Acceleration

      Acceleration (𝑎) is the rate of change of velocity. We can calculate the average acceleration using the following equation:

      \overline{a}=\frac{\Delta v}{\Delta t}=\frac{v_{f}-v_{i}}{t_{f}-t_{i}}

      Because velocity is a vector, acceleration is also a vector quantity. Instantaneous acceleration is acceleration measured at a specific instant in time. In most kinematic problems, we assume average acceleration is a constant value.
      • book icon
        Acceleration Book
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        Read this text. Pay attention to the examples which show how to solve equations of motion. These include how to calculate displacement, given average velocity and time, and how to calculate final velocity, given initial velocity, acceleration, and time.

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        More on Acceleration Page
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        This lecture accompanies what you just read.

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        Motion Equations for Constant Acceleration in One Dimension Book
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        Read this text for more examples and practice on how to solve motion equations for constant acceleration.

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        Average Acceleration Page
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        Watch this video to get a better understanding of acceleration and how it is graphed with respect to time.
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        Acceleration Equations Page
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        Watch this video to learn how to analyze acceleration with equations.

    • 2.6: Falling Objects

      Gravity is a force that attracts objects toward the center of the earth, or more generally speaking, massive objects to one another. In the absence of friction or air resistance, all objects fall with the same acceleration toward the center of the earth. This is known as free-fall. The acceleration due to gravity is g=9.80\frac{\mathrm{m}}{\mathrm{s}^{2}} .

      In reality, air resistance affects the acceleration of falling objects. Air resistance opposes the motion of an object in air, and causes falling lighter objects to accelerate less than heavier objects. This is why a feather falls to earth slower than a heavier object like a brick. If there was no air resistance, a feather and brick would fall to earth with the same acceleration due to gravity.
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        Falling Objects Page
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        Read this text for more about falling objects, gravity, and freefall. We will discuss how to calculate the examples here in the next section.

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        Vertical Motion in Free Fall Page
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        This lecture accompanies what you just read.

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        Zero Gravity Demonstration Page
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        Watch this live demonstration which takes place at zero gravity at the end of the last Apollo 15 moonwalk.

    • 2.7: Calculating the Kinematic Quantities of Objects in Constant Acceleration

      To perform calculations involving objects in constant acceleration situations, such as free fall, we first need to use the basic definitions of velocity and acceleration to derive useful formulas called "kinematic equations".

      We can use kinematic equations for any situation where there is a constant acceleration acting on an object (including zero acceleration), and included with this situation is freefall. In free fall, acceleration (a) equals the acceleration due to gravity (g). For an object falling, we use −g to show the vector's downward direction of free fall.

      • book icon
        Kinematic Equations for Objects in Free Fall Book
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        As you read, pay attention to the relevant equations in the box Kinematics Equations for Objects in Free Fall where Acceleration = −g

        v_{f}=v_{i}+a\Delta t

        y_{f}=y_{i}+v_{i}t+\frac{1}{2}a\Delta t^{2}

        v^{2}=v\tfrac{2}{i}+2a(y_{f}-y_{i})

        Note that because the motion is free fall, a is simply replaced with -g (here, g is the acceleration due to gravity, g=9.80\ \mathrm{m/s}^{2} ) and the direction of motion is the y direction, rather than the x direction. When calculating the position and velocity of an object in freefall, we need to consider two different conditions. First, the object can be thrown up as it enters freefall. For example, you could throw a baseball up and watch it fall back down.

        Complete the steps in Example 2.14. After you review the solution, pay attention to the graphs in Figure 2.40. You can throw an object directly downward as it enters freefall, such as when you throw a baseball directly down from a second-floor window.

        Then, complete the steps in Example 2.15. Notice that Figure 2.42 compares what is happening in Example 2.14 and Example 2.15. It is important to understand the difference between an object that is thrown up and enters free fall, versus an object that is directly thrown down. We can often use experimental data to calculate constants, such as g.

        In Example 2.16, we determine the acceleration due to gravity constant (g) from experimental data.
      • page icon
        Constant Acceleration Equations Page
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        Watch this video for details on how kinematic equations were derived.

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        More on Free Fall Page
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        Watch this video for details on kinematic equations and its relationship with freefall.

      • book icon
        Problem-Solving Basics for One-Dimensional Kinematics Book
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        Read this text for a step-by-step guide on how to solve problems of constant acceleration using kinematic equations.

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        Kinematic Equations in Constant Acceleration Page
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        This lecture accompanies what you just read.

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        Displacement with Constant Acceleration Page
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        This video goes into more detail about constant acceleration.

    • 2.8: Graphical Analysis

      When graphing two variables against each other, we generally define the dependent variable as the variable on the vertical axis (y-axis) and the independent variable as the variable on the horizontal axis (x-axis). When plotting a straight line, we use the equation y = mx + b , where m is the slope and b is the y−intercept of the line.

      We define slope as:

      m={\frac{rise}{run}}=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}

      The y−intercept is the point where the line crosses the y-axis of the graph.
      • book icon
        Graphical Analysis of One-Dimensional Motion Book
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        Read this text for an introduction on graphical position, velocity, and acceleration with regards to one another. As you read, pay attention to Figure 2.46 which is an example of a linear graph is the graph of position versus time when acceleration is zero.

        See an example of this type of graph in Figure 2.47. In this graph, we can determine the slope by picking two different points on the line, taking the change in y-value, and dividing it by the change in x-value between those two points. In this case, the unit for slope is m/s, which is the unit for velocity. Therefore, the slope for a graph of position versus time with zero acceleration is the average velocity of that object. See how to calculate the average velocity of an object from this type of graph in Example 2.17.

        When acceleration is a non-zero constant, the graph of position versus time is no longer linear. You can see an example of this type of graph in Figure 2.48. Note that while the position versus time graph is not linear, the velocity versus time graph is linear. In the position versus time graph, the slope at any given point is the instantaneous velocity of the object. The instantaneous slope can be determined by drawing a tangent line at the desired point along the graph and determining slope. Pay attention to the tangent lines drawn in Figure 2.48 (a).

        To determine instantaneous velocity at a given time when acceleration is a non-zero constant, take a look at Example 2.18. We can determine instantaneous velocities at multiple points along a position-time graph with constant non-zero acceleration and make a table relating these instantaneous velocities to the specified time along the x-axis where we found them. Then, we can use that table to plot velocity versus time. This process is demonstrated in Figure 2.48 (a) and (b). The slope of this linear graph has units m/s^{2} , which are acceleration units. Therefore, the slope of the velocity versus time graph is acceleration.
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        Interpreting Velocity Graphs Page
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        Watch this video for details regarding the velocity versus time graphs and how they relate to acceleration.

    • Unit 2 Assessment

      • quiz icon
        Unit 2 Assessment Quiz
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        Take this assessment to see how well you understood this unit.

        • This assessment does not count towards your grade. It is just for practice!
        • You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
        • You can take this assessment as many times as you want, whenever you want.

  • Unit 3: Kinematics in Two Dimensions

    Most motion in nature follows curved paths rather than straight lines. Motion along a curved path on a flat surface or a plane is two-dimensional and thus described by two-dimensional kinematics. Two-dimensional kinematics is a simple extension of the one-dimensional kinematics covered in the previous unit. This simple extension will allow us to apply physics to many more situations and it will also yield unexpected insights into nature.

    Completing this unit should take you approximately 5 hours.

    • Upon successful completion of this unit, you will be able to:

      • add and subtract vectors;
      • determine the components of a vector given its magnitude and direction and determine the magnitude and direction of a vector given its components; and
      • separately analyze the horizontal and vertical motions in projectile problems.

    • 3.1: Introduction to Kinematics in Two Dimensions using Vectors

      Two-dimensional kinematics is surprisingly easy. They are similar to one-dimensional problems, due to our coordinate system. Notice that coordinate systems have perpendicular axes, and motion along the two axes is independent from each other. So, the physics or math that helps us solve for an object's motion in the x−direction does not influence its motion in the y−direction. Solving for two-dimensional motion is like solving for one-dimensional motion twice!

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        Kinematics in Two Dimensions Page
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        Read this text, which provides a general overview of the concept of kinematics in two dimensions.

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        Unit Vectors and Engineering Notation Page
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        To solve two-dimensional kinematic problems, we first need to understand how two-dimensional motion is represented and how to break it up into two one-dimensional components. We also need to understand vectors. A vector is a quantity that has both a magnitude (amount) and direction. Often in texts, vectors are denoted by being bolded or having a small arrow written above the vector name.

        For example, a vector called A can be written as A or as \overrightarrow{A} . The magnitude, or amount, of the vector A equals the value of A. And, the direction of A is given by some other notation usually accompanying the value. We can think of vectors as arrows, with the length being the magnitude of the vector and the arrow pointing in the direction of the vector.

        Vectors are often notated like this: \vec{A}=A_{x}\hat{x}+A_{y}\hat{y} . The \hat{x} denotes that the magnitude A_{x} is the part of the vector that protrudes the x-axis. Similarly, the \hat{y} denotes that the magnitude A_{y} is the part of the vector that protrudes the y-axis.

        So, for example, the vector \vec{A}=3\hat{x}+5\hat{y} extends down the x-axis three units while extending up the y-axis five units. This notation is called "component form" and is a preferred way of representing vectors.

        Another way of representing vectors is by denoting their magnitude and direction. For example, we can denote the vector A, shown in the previous paragraph, also as 5.83 units 59 degrees from the x-axis. Notice that we need to specify that the direction has an angle with respect to the x-axis. Not only do we need an angle, but we also need a reference point from which the angle spawns. Generally, the x-axis is a convenient choice. We call this notation the magnitude-direction form.

        This video discusses vector notation. Note that they use engineering notation, which replaces x-hat with i-hat and y-hat with j-hat. The meanings are the same despite these changes.

    • 3.2: Adding and Subtracting Vectors

      When adding or subtracting vectors, we can follow many of the rules we learned in math class about non-vector numbers. Vector addition follows the commutative property, which means the order of addition does not matter. Vector addition also follows the associative property, which means it does not matter which vector is first when vectors are being added.

      One way to add or subtract vectors is to do so graphically. The graphical method for adding and subtracting vectors is called the head-to-tail method. When adding vectors using this method, use these steps:

      1. Draw the first vector starting from the tail, or starting point of the vector, to its head, or ending point (arrow) of the vector.
      2. Begin the second vector by putting its tail at the head of the first vector.
      3. Finally, draw a line from the tail of the first vector to the head of the second vector.

      The vector that results is the resultant vector, or the solution to the vector addition problem. To determine the magnitude of the resultant vector, measure it with a ruler. To determine the direction of the resultant vector, use a protractor to determine the angle from one of the axes. When subtracting vectors graphically, consider the vector that is being subtracted as negative. That means the direction of the vector being subtracted is flipped so it points in the opposite direction. The head-to-tail process is the same as it is for addition.

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        Vector Addition and Subtraction Page
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        As you read, pay attention to the worked examples: using the head-to-tail method to add multiple vectors in Example 3.1 and using the head-to-tail method to subtract vectors in Example 3.2.

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        Vector Addition Using the Graphical Method Page
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        This lecture accompanies what you just read.

    • 3.3. Adding Vectors Analytically: Determining the Components, Magnitude, and Direction of a Vector

      We can also use analytical methods to add and subtract vectors. Analytical methods use trigonometry to solve vector addition and subtraction. While we still use arrows to represent vectors, analytical methods reduce the measurement errors that can occur with graphical (head-to-tail) methods.

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        Analytical Methods for Vector Addition and Subtraction Page
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        Read this text, which explains how we need to resolve vectors into their component vectors in the x-y coordinate system when using analytical methods to solve vector problems. See Figure 3.26 for an example of a vector that has been resolved into its x and y components. Here, the vector A has a magnitude A and an angle 𝛳. We can break the vector down into two components: Ax and Ay. We know that Ax + Ay = A . However, we must use trigonometry to determine how the scalar or magnitude part of each vector relates to one another. You do not need to know the inner workings of trigonometry to deal with vectors analytically, but you need to understand their basic functions and know how to input a sine and cosine function into a calculator. The magnitudes of the component vectors relate to the resultant vector this way:

        A_x=A\cos\theta

        A_y=A\sin\theta

        See Figure 3.27 for an example of a vector that has been resolved into its component vectors and shows the magnitudes of the component vectors. Note that these equations work if you want to find the angle at the bottom-left of the right triangle in Figure 3.27. If you want to find the angle at the top of the triangle, you would use the sine function for A_{x} and the cosine function for A_{y} .

        The general rule is: use the sine function for components of the vector that are opposite of the triangle as the angle, and use the cosine function for components of the vector that are adjacent on the triangle to the angle. For example, in Figure 3.27, you can see that the y-component of the vector is on the opposite side of the triangle as the angle \theta so the sine function is used to find the y-component, as in the previous equation.

        Sometimes you are given the component vectors and need to determine the magnitude and angle of the resultant vector. To do this, we again use the trigonometry of right triangles:

        A=\sqrt{A_x^2+A_y^2}

        \theta=\tan^{-1}\left( \frac{A_y}{A_x} \right )

        The angle obtained by using the tangent equation is such that the opposite component of the vector is the y-component, and the adjacent component is the x-component. Also, pay attention to the example of a resultant vector calculated from its component vectors in Figure 3.29.
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        More on Vector Addition Page
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        This lecture accompanies what you just read.

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        Vector Components on a Grid Page
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        This video discusses how vectors are represented as components in the x- and y-axes.

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        Projectiles at an Angle Page
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        This video gives an example of how we use components representing vectors in the x- and y-axes in two-dimensional kinematic problems.

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        Visualizing Vectors in Two Dimensions Page
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        This video describes how to convert vectors from magnitude-direction form into component form, and vice versa.

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        Unit Vector Notation Page
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        This video demonstrates how to add vectors using the graphical and analytical methods.

    • 3.4: Projectile Motion and Trajectory

      Often, physics problems occur on the surface of the Earth, such as footballs being kicked, rockets being fired, and daredevils riding their motorcycles off cliffs. This means that the y-component of these two-dimensional motions involve acceleration pointing downward while the x-component does not have any acceleration. We call these types of motion projectile motion.

      We define projectile motion as the motion of a thrown object that only feels the acceleration of gravity. The projectile is the object being thrown; the trajectory is the path the object takes when it is thrown.

      We need to use the kinematic equations we learned in Unit 2 of this course to calculate projectile motion, for each of the two-dimensions separately. Note that we assume there is no air resistance when we perform projectile motion calculations – so gravity is the only force acting on the projectile.

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        Projectile Motion Page
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        This reading discusses how vectors are represented as components in the x- and y-axes.

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        More on Projectile Motion Page
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        This lecture accompanies what you just read.

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        Another Way to Determine Time In Air Page
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        Watch this video to see another way to solve for the time of a projectile in air. Since time transcends the x- and y-components of a trajectory, it is important to know how to calculate time so you can use it to connect motion in the two dimensions.

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        Horizontally-Launched Projectiles Page
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        Watch this video to learn how to solve for a horizontally-launched projectile, with no initial y-component to velocity.

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        Launching and Landing at Different Elevations Page
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        Watch this video on how to solve kinematic equations for the x and y directions of motion, using the same procedures as for a horizontally-launched projectile.

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        Total Displacement for a Projectile Page
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        Watch this video for another example of projectile motion and how to solve for quantities using kinematic equations for both dimensions.

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        Total Final Velocity for a Projectile Page
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        Watch these two videos on how to solve for the total final velocity of the projectile at the end of its path. Note that the presenter makes a small mistake which they correct in the second video.

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        Projectiles on an Incline Page
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        Watch this video on how to solve projectile motion problems using kinematic equations for trajectories that start at an incline.

    • Unit 3 Assessment

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        Unit 3 Assessment Quiz
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        Take this assessment to see how well you understood this unit.

        • This assessment does not count towards your grade. It is just for practice!
        • You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
        • You can take this assessment as many times as you want, whenever you want.

  • Unit 4: Dynamics

    Kinematics is the study of motion. It describes the way objects move, their velocity, and their acceleration. Dynamics consider the forces that affect the motion of moving objects. Newton's Laws of Motion are the foundation of classical dynamics. These laws provide examples of the breadth and simplicity of principles under which nature functions.

    Completing this unit should take you approximately 8 hours.

    • Upon successful completion of this unit, you will be able to:

      • compare and contrast mass and inertia;
      • determine the net force on an object;
      • draw and interpret free-body diagrams representing the forces on an object;
      • identify the correct use of normal and tension forces in terms of Newton's Third Law of Motion;
      • use Newton's Second Law of Motion to analyze dynamic problems;
      • apply Newton's Law of Gravity; and
      • give examples of the effects of friction on the motion of an object.

    • 4.1: Newton's First Law of Motion

      Newton's First Law of Motion is also called the Law of Inertia: an object at rest will remain at rest unless an outside force acts upon it. Also, an object in motion with constant velocity will remain in motion with constant velocity unless an outside force acts upon it. The reason for this law is to exemplify the functions of mass.

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        Newton's First Law of Motion and Inertia Page
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        Read this text, which explains how we define mass as the amount of matter in an object. We measure mass in units, such as kilograms. Mass does not depend on the strength of the gravitational field and therefore does not depend on the location where it is being measured.

        Inertia is the property of matter which mass quantifies. It describes the fact that an object at rest (not moving) will stay at rest unless an outside force acts upon it. Likewise, an object in motion will stay in motion with constant velocity unless an outside force acts upon it. These outside forces accelerate the object.

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        Force and Newton's Laws Page
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        This lecture accompanies what you just read.

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        The Historical Context Page
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        This video provides a brief explanation on the thought process behind Newton's First Law and some historical context to the Law of Inertia.

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        More on Newton's First Law of Motion Page
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        This video provides a more detailed explanation behind the concepts of Newton's First Law.

    • 4.2: Newton's Second Law of Motion

      As we saw in Newton's First Law of Motion, an object at rest stays at rest unless acted upon by an external force. Also, an object in motion at constant velocity remains in motion unless acted upon by an external force. Again, this is inertia. The only way to overcome inertia is to accelerate the object. Applying a net force to the system to induce acceleration.

      Acceleration is proportional to the net external force on a system. That is, the higher the applied force, the bigger the acceleration. We also know that acceleration is inversely proportional to mass. That is, large objects accelerate at a slower rate than smaller objects. We know this from our everyday observations. It is easier to accelerate a light ball than a heavier bowling ball.

      Newton's Second Law of Motion relates net external force to acceleration and mass of the system: F_{\mathrm{net}} = ma , where F_{\mathrm{net}} is the net force, m is mass, and a is acceleration. Note that force is a vector quantity, so it has a magnitude and a direction.

      The system is whatever we are interested in when calculating a physics problem. The external force is any force that acts upon the system, but is not part of the system. For example, picture pushing a rock up a hill. The system is the rock, and the external force is you pushing the rock.

      The unit for force is the Newton, N. The definition of the Newton is 1\ \mathrm{N} = 1\ \mathrm{kg\: m/s^{2}} .
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        Newton's Second Law of Motion Page
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        This video reviews how Newton's Second Law was derived using concepts that are familiar from previous units.

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        Using Newton's Second Law Page
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        As you read, pay attention to examples 4.1 and 4.2, which use Newton's Second Law of Motion to calculate acceleration and force in objects in motion.

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        F=ma Page
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        This lecture accompanies what you just read.

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        Examples of Newton's Second Law Page
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        This video discusses Newton's Second Law and solves a few sample problems. It delves into a more detailed analysis of solving for Force, Mass, and Acceleration of dynamic situations.

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        This video presents additional sample problems involving Newton's Second Law.

    • 4.3: Free-Body Diagrams

      A free-body diagram shows all of the forces acting upon a system. It is a simplified way to visualize what is happening during a physics problem. Drawing the forces as vector arrows in the direction of the force from the center of the system can help us figure out how we need to add or subtract force vectors when determining the net force on an object.

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        The Concept of Force Page
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        Read this text to see examples of how to draw a free-body diagram like we saw earlier in this unit. The text also discusses force as a vector and introduces a way to visualize multiple forces acting on an object: the free-body diagram. Notice the free-body diagrams drawn for specific examples in Figures 4.5 and 4.6.

    • 4.4: Newton's Third Law of Motion

      Newton's Third Law of Motion states that for every force exerted by an object to another object, there is an equal magnitude force exerted on the first object by the second object in the opposite direction. Some call this the Law of Action and Reaction.

      In other words, for every action (exerted force), there must be an equal and opposite reaction. This law tells us that forces are always paired. Keep in mind that these forces act on two separate objects in pairs. The forces do not act on the same object being pushed.

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        Symmetry in Forces Page
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        As you read, pay attention to the example which applies Newton's Third Law of Motion to a swimmer in a pool in Figure 4.9. When the swimmer kicks off the wall of the pool to begin swimming, the swimmer exerts a force toward the wall.

        Because of the Third Law, the wall also exerts an opposing force back on the swimmer. The force by the wall on the swimmer is equal in magnitude, but opposite in direction of the force exerted by the swimmer on the wall. In the other axis, gravity exerts a force toward the earth on the swimmer, but interestingly enough, the swimmer is also exerting an equal amount of force on the Earth pulling it up toward them. These are both examples of Newton's Third Law in action. See another example of determining the forces in a given system in Example 4.3.

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        Newton's Third Law and F=ma Page
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        This lecture accompanies what you just read.

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        More on Newton's Third Law Page
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        Watch this video for another presentation of Newton's Third Law.

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        Examples of Newton's Third Law Page
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        Watch these two videos for examples of action-reaction pairs of forces applied between two objects in contact with each other. They will help you solve problems using Newton's Third Law.

    • 4.5: Solving Problems Using Newton's Second Law: Weight

      There are four common classical forces that will be discussed in the following sections: weight, normal force, tension, and friction. We will discuss each of these forces one at-a-time in each of the following sections.

      Weight refers to the force of gravity on an object of a given mass. Because it comes from gravity, the weight force is generally directed toward the earth. The equation that relates the mass of an object to its weight is F_{g}=mg . This equation works only on or near the surface of the Earth.

      Let's consider a coffee cup sitting on a table. The coffee cup is experiencing the force of its weight that draws it toward the center of the earth. This "pushes" down on the table. However, because of Newton's Third Law of Motion, there must be an equal magnitude force in the opposite direction also acting on this table to balance the forces.
      • page icon
        Problem-Solving Strategies Page
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        As you read, note that we can use Newton's Second Law of Motion to solve problems that involve forces. You should follow the following four steps when solving these types of problems.

        1. Sketch the system described in the problem.
        2. Identify forces and draw the forces on the sketch.
        3. Draw a free-body diagram of the forces acting on the system.
        4. Use Newton's Second Law of Motion to solve the problem.

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        Further Applications of Newton's Laws of Motion Page
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        As you read this text pay attention to the worked examples of how to solve dynamics problems using the strategies we discussed previously. See examples 4.7, 4.8, and 4.9. We will discuss each of the forces involved soon.

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        Identifying and Labeling Types of Forces Page
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        Watch this video from 7:24 to 8:35 for a brief explanation of the force due to gravity that we call weight.

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        Mass and Weight Page
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        What is the difference between mass and weight? This video will go into the differences of the two concepts and situations where knowing these differences will be useful.

      • page icon
        Characteristics of Forces Page
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        This video demonstrates the characteristics of the forces of tension, friction, weight, and normal when solving problems related to forces.

      • page icon
        Gravity and Weight Page
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        This video explores gravity, one of the fundamental forces. The narrator explains gravitational interactions in terms of the gravitational field and describes when flat-earth-gravity is a valid approximation.

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        Resultant Forces Page
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        This video explores types of forces: normal contact force, tension, friction, air resistance, magnetic force, electrostatic force, and gravitational force. It explains that force is a push or a pull that acts on an object.

        Forces are vector quantities because they have both magnitude and direction, and so can be represented by an arrow. Scalar quantities have only magnitude and no direction. When several forces act on an object they can be replaced by a single force that has the same effect. This single force is called the resultant force.

    • 4.6: Newton's Law of Gravity

      Weight, in a more general sense, can be given using Newton's Universal Law of Gravitation. This law states that all objects in the universe attract each other in straight force lines between them.

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        Newton's Universal Law of Gravitation Page
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        Read this text to see an example of how two objects exert gravitational forces on each other in a straight line in Figure 6.21. The force between two objects is directly related to the product of the masses and is inversely proportional to the distance between the objects squared. For two objects with masses M and m and radius r , we can write this as F_{g} = G\frac{Mm}{r^2} , where G is the gravitational constant, 6.674\times 10^{-11} {\frac{\mathrm{Nm^2}}{\mathrm{kg^2}}}

        As you can see from the formula, distance plays a large role in the gravitational force acting between two masses. If two masses feel an initial attractive force due to gravity, and then become twice as far from each other, they will now experience a quarter the force as before.
      • page icon
        Introduction to Gravity Page
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        This video introduces the concept of the Universal Law of Gravitation and relates it to its local law version ( F=mg ).
      • page icon
        Introduction to Newton's Law of Gravitation Page
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        Watch these two videos for a demonstration of using the Universal Law of Gravitation for finding the local acceleration on Earth's surface and of a space station near Earth.

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        Gravity for Astronauts in Orbit Page
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        Watch this video for detailed analysis of gravity as it applies to astronauts flying high above the atmosphere.

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        Would a Brick or Feather Fall Faster? Page
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        Watch this video to analyze why things fall at the same rate. It uses the Universal Law of Gravitation to prove constant acceleration on Earth for all objects.

    • 4.7: Solving Problems Using Newton's Second Law: Normal Force

      Otherwise known as the "force of contact", we have the normal force. Here, "normal" essentially means perpendicular. That is, we give it the name normal force to make it obvious that it points perpendicularly to a surface.

      For example, normal force balances the weight from a cup of coffee on a table and keeps it from going through the table. This is why we often call normal force the force of contact. In other words, we can say that normal force is the force that prevents two objects from being in the same place. The normal force is often abbreviated as N. Do not confuse the symbol for normal force as the unit of force, Newtons.

      • page icon
        Normal Force Page
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        Read this section, which discusses normal force.

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        Identifying and Labeling Types of Forces Page
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        Watch this video until 4:37.

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        More on Normal Force Page
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        This video explains normal force: the force of contact. Again, "normal" essentially means perpendicular. The opposing force is the normal force.

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        Normal Force and Contact Force Page
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        This video introduces the concept of normal force and compares it to the weight of an object.

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        Normal Force in an Elevator Page
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        This video discusses the concept of normal force as it deviates from being exactly equal to the weight of an object, such as inside an accelerating elevator.

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        Multiple Forces Alongside Normal Force Page
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        These two videos describe more complicated situations dealing with multiple forces (some going diagonal) accompanying normal force.

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        Ice Accelerating Down An Incline Page
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        This video puts the forces we have discussed together, as in the situation of an inclined plane.

    • 4.8: Solving Problems Using Newton's Second Law: Tension

      Tension is the force along the length of an object. We normally think of tension as a force of the object's strength, such as for a rope. Objects, such as ropes, can only exert forces in the same direction as their length. If a rope is attached to a hanging object, the object's weight exerts a force toward the earth while the rope acts as a tension force in the opposite direction of the weight. Much like the normal force from the table mentioned earlier, this keeps the object from falling down.

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        Tension Page
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        As you read, pay attention to the example of tension in Figure 4.15 as it talks about how tension is distributed along a rope carrying a weight.

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        Identifying and Labeling Types of Forces Page
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        Watch this video from 6:34 to 7:03.

      • page icon
        Tension Forces Page
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        This video explains how tension forces are directed in a simple situation and how it fits in with other classical forces.

    • 4.9: Solving Problems Using Newton's Second Law: Friction

      Friction is the force between two surfaces in contact that opposes parallel motion between them. Kinetic friction is the friction between surfaces that are moving or sliding relative to each other. Static friction is the friction that occurs that prevents two surfaces from moving or sliding with respect to each other.

      Static friction varies based on how much counter force is needed to prevent two objects from sliding. When a certain amount of force is applied to a stationary object in contact with a surface, static friction serves to counter that applied force in order to keep the object and surface in contact from sliding. The more force that is applied, the more static friction is summoned to counter it.

      However, static friction is only so strong. So once the maximum static frictional force is summoned, the object will start to slide and the static friction force will give way to kinetic friction force. Generally, the maximum ability of static friction is higher than the maximum ability of kinetic friction. That is, once the maximum static friction force is met, the object undergoing applied force will jolt forward because the kinetic friction that took over is weaker than the maximum static friction force that held it previously.

      Kinetic and static frictional forces are given by the equations:

      F_{fr,k}=\mu _{k}F_{N}
      F_{fr,s}\leq \mu _{s}F_{N}

      Note that the static friction force equation is an inequality. That is because static friction only aims to counter potential movement or sliding between two surfaces, which vary based on the magnitude of the applied force. Both equations have a \mu symbol which is called the coefficient of friction and depends solely on the types of materials that make up the two surfaces. For example, between steel and ice, the coefficient of friction would be very small (perhaps 0.1). However, between rubber and concrete, the coefficient of friction would be rather large (perhaps 0.8). The coefficient of friction is rarely larger than one.

      We experience friction often in our everyday lives. For example, if you slide a box across a room, the box's motion will eventually stop due to the friction that occurs between the surface of the box and the surface of the floor. A box will slide relatively well across a smooth tile floor because the smooth tile floor provides a lower frictional force. It will slide less well across a floor with a rough carpet because the carpet provides a higher frictional force.

      When we walk on a sidewalk our shoes do not generally slip because the static friction between our shoes and the sidewalk opposes the forward force of our shoes. However, we know that icy surfaces are "slippery" when the ice exerts less friction on our shoes than concrete.
      • page icon
        Identifying and Labeling Types of Forces Page
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        Watch this video from 4:37 to 6:34.

      • page icon
        Friction Page
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        The text explains the fundamentals of friction that we discussed earlier, but in more detail.

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        Kinetic and Static Friction Forces Page
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        This video demonstrates the difference between kinetic and static friction.

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        Friction and Force from Springs Page
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        This lecture accompanies what you just read. It talks about the equations of kinetic and static friction forces, and the concepts behind friction itself. Watch the video until the 6:12 mark, where Clements begins discussing springs and other material we will cover in another Saylor course, PHYS102 Introduction to Electromagnetism.

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        Comparing Static and Kinetic Friction Page
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        Watch this video to learn about the differences between static and kinetic friction, and why maximum static friction is generally stronger than kinetic friction.

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        Examples of Static and Kinetic Friction Page
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        This video explains how to solve basic friction problems involving static and kinetic cases.

    • Unit 4 Assessment

      • quiz icon
        Unit 4 Assessment Quiz
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        Receive a grade

        Take this assessment to see how well you understood this unit.

        • This assessment does not count towards your grade. It is just for practice!
        • You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
        • You can take this assessment as many times as you want, whenever you want.

  • Unit 5: Rotational Kinematics

    Now that we have discussed forces and how they manipulate motion, we will begin exploring a particular force that makes objects move in a curved motion. In this unit, we study the simplest form of curved motion: uniform circular motion, or motion in a circular path at a constant speed. In some ways, this unit is a continuation of the previous unit on dynamics, but we will introduce new concepts such as angular velocity and acceleration, centripetal force, and the force of gravity.

    Completing this unit should take you approximately 3 hours.

    • Upon successful completion of this unit, you will be able to:

      • explain why an object moving at a constant speed in a circle is accelerating;
      • solve problems involving planets and satellites;
      • explain what it means when an astronaut orbiting the Earth is described as being weightless;
      • compare and contrast the physical properties associated with linear motion and rotational motion;
      • state and explain Newton's Law of Universal Gravitation; and
      • use kinematic equations to solve circular motion problems.

    • 5.1: Centripetal Force

      A centripetal force is any force that makes an object move in a circular motion. A centripetal force can involve any of the classical or electromagnetic forces, such as gravitational force (weight), normal force, tension, and friction.

      For example, gravitational force acts as a centripetal force on a planetary scale because it causes planets to orbit in a circle. However, gravitational force is not a centripetal force on the Earth's surface because gravity makes objects fall straight down toward the Earth's center, not in a circle. A normal force can act as a centripetal force, such when a roller coaster does a loop-da-loop. Friction can act as a centripetal force, such as when it causes cars to turn corners on a road.

      The equation for centripetal acceleration is a_c=\frac{v^2}{r} , where v is object speed and r is radius (distance from center).
      • page icon
        Centripetal Acceleration Page
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        Read this text, which presents more explanation on the topic of centripetal acceleration. Pay attention to Figure 6.8 which shows an example of centripetal acceleration. In this example, a disk is rotating at a constant speed. As the disk rotates, the velocity vector at any given point on the disk changes because the direction changes. As shown in the free-body diagram at the top of the figure, the velocity vectors add to make a net velocity vector toward the center of the disk. This leads to centripetal acceleration because there is a net change in acceleration toward the disk.

        Centripetal forces assume the equation F_c=m\dfrac{v^2}{r} . Therefore, whatever equation characterized the classical forces in Unit 4 can also be related to a situation by its centripetal force equation just given.
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        Circular Motion and Centripetal Acceleration Page
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        This lecture accompanies what you just read.

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        Centripetal Force Page
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        Read this text, which demonstrates using the centripetal force equation in conjunction with the classical forces we learned in the previous unit: weight, normal force, tension, and friction.
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        Tennis Ball on a String Page
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        Watch this video for a demonstration of centripetal force.
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        More on Centripetal Force Page
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        This lecture accompanies what you just read.

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        Visualizing Centripetal Acceleration Page
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        Watch this video to see how the equation for centripetal force is derived.

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        More on Centripetal Force And Acceleration Page
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        Watch this video as it goes into the concept of centripetal acceleration and centripetal force.

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        Example: Loop de Loop Page
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        Watch these two videos for examples of how to use normal force and gravity as a centripetal force in a loop-da-loop problem.

      • 5.2: Centripetal Force and the Universal Law of Gravitation

        A dramatic application of centripetal force is the Universal Law of Gravitation. Johannes Kepler (1571–1630), the German astronomer and mathematician, created three laws that pertain to orbital motion during the Renaissance period. At this time, physics and astronomy were two separate fields of study. Kepler developed these three laws independently from the laws of physics we know today.

        • book icon
          Satellites and Kepler's Laws Book
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          Read this text, which includes visual diagrams of Kepler's Laws of Planetary Motion, which describe the motion of planets around the sun. We can also apply these laws to explain the motion of satellites around planets.

          1. Kepler's First Law of Planetary Motion states that planets move around the sun in an ellipse shaped orbit with the sun at the center of the ellipse (see Figure 6.29).
          2. Kepler's Second Law of Planetary Motion states that planets move so that a point on the planet sweeps an equal area in equal times (see Figure 6.30).
          3. Kepler's Third Law of Planetary Motion refers to the relationship between the time it takes for two planets to revolve around the sun, and their distances from the sun: \frac{T_1^2}{T_2^2}=\frac{r_1^3}{r_2^3} , where T_{1} and T_{2} are periods of orbit while r_{1} and r_{2} are radii for planets one and two.

          We can use Kepler's Third Law to solve problems to determine the period for planetary or satellite orbits. See a worked example of using the equation from Kepler's Third Law to determine the period of a satellite in Example 6.7. Pay attention to the derivation of Kepler's Third Law using the concept of centripetal forces.
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          Kepler's Three Laws of Planetary Motion Page
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          This video accompanies the text on Kepler's Three Laws.

    • 5.3: Angular Position, Velocity, and Acceleration

      In rotational motion, we deal with two-dimensional motion. Unlike with linear motion, we need to define angles and distances associated with circular motion.

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        Read this text. To understand circular or rotational motion, picture a spinning disk, such as the picture of a CD in Figure 6.2. This figure shows a CD with a line drawn from the center to the edge. All of the points along this line travel the same angle, in the same amount of time, as the CD spins. We call this the rotational angle, which is defined as \Delta \theta=\frac{\Delta s}{r} . We call the distance along the circumference traveled ( \Delta s ) the arc length, and we call the radius of the circular motion ( r ) the radius of curvature.

        When describing angles, we often use the unit radian, abbreviated as rad. We define radians as 1 revolution = 2\pi rad. Radians are the standard unit for physics problems, but we can convert radians to the more familiar degrees for convenience. Pay attention to Table 6.1 for conversions between radians and degrees.

        We define angular velocity (or rotational velocity), \omega (the Greek letter omega), as the rate at which the angle changes while an object is rotating. We can write it as \omega=\frac{\Delta \theta}{\Delta t} , where \Delta \theta is the change in angle and \Delta t is the time it takes for the angle to change that amount.

        We can relate angular velocity to linear velocity using the relation v = r \omega , with r being the radius of curvature. Pay attention to the derivation of how angular velocity relates to linear velocity in equations 6.6, 6.7, 6.8, and 6.9.
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        We define angular acceleration as the change in angular velocity with respect to time. The equation is \alpha = \frac{\Delta \omega}{\Delta t} , where \alpha represents angular acceleration.

        As you read, pay attention to Example 10.1, which shows how to calculate the angular acceleration of a bike wheel. In the first part of the problem, we calculate the angular acceleration of the wheel given the change in angular velocity and time. In the second part of the problem, we calculate the time needed to stop an already spinning wheel given angular deceleration as initial velocity, using the same angular acceleration equation. See a diagram of a rotating object showing the relationship between linear and angular velocity in Figure 10.3.

    • 5.4: Kinematics of Rotational Motion

      Rotational Kinematics is the study of rotational motion, much like linear kinematics (or just plain kinematics) is the study of linear motion. When solving kinematics problems of rotational motion, we look at the relationships between angular and linear versions of position, velocity, and acceleration.

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        Read this text to see additional worked examples of how to solve problems involving the kinematics of rotational motion.

        Example 10.3 shows how to calculate the kinematics of an accelerating fishing reel. Here, equation 10.19 is used to determine how the angular velocity changes with time. This result is used to calculate linear speed. Example 10.4 is an example where the fishing reel decelerates. Using equation 10.19, we solve for time rather than angular velocity. To see more of these types of problems, review Examples 10.5 and 10.6.

    • Unit 5 Assessment

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        Unit 5 Assessment Quiz
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        Take this assessment to see how well you understood this unit.

        • This assessment does not count towards your grade. It is just for practice!
        • You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
        • You can take this assessment as many times as you want, whenever you want.

  • Unit 6: Rotational Statics and Dynamics

    What do desks, bridges, buildings, trees, and mountains have in common – at least in the eyes of a physicist? The answer is that they are ordinarily motionless relative to the Earth. Consequently, their acceleration, with respect to the Earth as a frame of reference, is zero. Newton's second law states that net F = ma, so the net external force is zero on all stationary objects and for all objects moving at constant velocity. There are forces acting, but they are balanced. That is, the forces are in equilibrium.

    Completing this unit should take you approximately 1 hour.

    • Upon successful completion of this unit, you will be able to:

      • define the conditions necessary for a rigid body to be in equilibrium;
      • define the vector quantity torque;
      • define rotational inertia;
      • compare and contrast the dynamics of linear and rotational motion;
      • Solve statics problems with rotational motion; and
      • solve dynamics problems involving rotational motion.

    • 6.1: Conditions for Equilibrium

      When an object is in equilibrium, the forces acting upon the object are balanced. That is, the net force on the object is zero. For this to occur, the object must either not be moving, or it must be moving at a constant velocity.

      There are two types of equilibrium: static equilibrium and dynamic equilibrium.

      1. Static equilibrium describes a system that is balanced and does not rotate. An example is a seesaw where two children sitting at either end are exactly the same weight. Since the seesaw will not move, it is in static equilibrium.

      2. Dynamic equilibrium describes a system that is balanced, but also moving (without any angular acceleration). An example is a planet in perfect circular orbit around its parent star. There is no torque acting on the planet making it orbit faster or slower, but it will keep orbiting for a very long time. The system is in equilibrium, and because it is moving, it is dynamic.

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        The First Condition for Equilibrium Page
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        As you read, pay attention to the illustration of static equilibrium in Figure 9.3 and the illustration of dynamic equilibrium in Figure 9.4. An object in static equilibrium is completely motionless. An object in dynamic equilibrium is moving at constant velocity.

        The study of statics is the study of objects that are in equilibrium. Two important conditions must be met for an object to be in equilibrium. First, the net force on the object must be zero. Secondly, a rotating object does not experience rotational acceleration. That is, a rotating object can be in equilibrium if its rotational velocity does not change.

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        Watch this video, which accompanies what you just read.

    • 6.2: Torque

      A system is not in equilibrium when a rotational force is acting on it to make it accelerate in its rotation. We call this rotational force torque. We define torque as the force to turn or twist an object, thus changing its rotational velocity. The unit for torque is the Newton-meter (Nm).

      We can write the definition of torque as \tau = rF\sin\theta , where \tau (the Greek letter tau) is torque, r is how far the force is applied from the axis of rotation, F is the force magnitude, and \theta is the angle between the force and radial vector from the axis of rotation and where the force is being applied.
      • book icon
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        As you read, pay attention to the diagram of an object rotating on a frictionless table in Figure 10.11. We can see the radius from the center of the table (the pivot point) and the mass at the end of the radius.

        When solving dynamics problems, we first need to identify the system and draw a free-body diagram of all the forces acting upon the system. Once the forces acting upon the system are defined, we can use the torque equation and angular acceleration equations to solve the problem: \tau_{net}=I\alpha , where I is the moment of inertia, \tau is torque, and \alpha is the rotational acceleration due to the torque.

        Example 10.7 shows how to use these equations to determine the angular acceleration of a person pushing a merry-go-round. Here, the first step is to calculate torque. The next step is to calculate the moment of inertia. Finally, torque and moment of inertia are used to calculate the angular acceleration on the merry-go-round.
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        Rotational Kinematics and Dynamics Page
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        Watch this video, which accompanies what you just read. Note that Greg Clements discusses the moment of inertia or rotational inertia, and Figure 9.6 which is in our next reading.

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        The Second Condition for Equilibrium Page
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        We define the moment of inertia, or rotational inertia, as mr^{2} for a point mass, where m is the mass of the object being rotated and r is the radius from the pivot point to the end of the mass.

        However, the moment of inertia for a distribution of mass that makes a shape, such as a rotating cylinder or sphere, applies different equations. For example, for a solid sphere rotating about a central axis going through the core of the sphere, the moment of inertia is I_{\mathrm{solid\ sphere}}=\frac{2}{3}mr^{2} where m is the mass of the entire sphere and r is the sphere's radius.

        In Section 6.1, we said that an object in equilibrium must have no rotational acceleration. We can restate this by saying that an object in equilibrium must have a torque of zero. When no torque is acting on a system, no rotational acceleration is given to the system, and it remains in equilibrium.

        As you read, pay attention to Figure 9.6, which shows the torque on a rotating plank of wood secured at a pivot point at one end. This diagram shows how the direction of the force impacts the rotation of the plank of wood.

        When the force is perpendicular to the length of the plank of wood, the plank experiences torque, and it rotates. When the force is parallel to the length of the plank of wood, it does not experience a net force and therefore does not rotate or experience torque. When the force is at an angle other than 90° from the length of the plank, the plank experiences less torque than if the force was at 90° from the plank's length.
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        Moment of Inertia Page
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        This video discusses the concept of moment of inertia.
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        The Race Between a Ring and a Disc Page
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        This video offers demonstrations of rotational inertia, the property of an object that deals with the resistance to a change in the state of rotational motion. This depends on the mass of the object and the way that mass is distributed from the axis of rotation.

    • 6.3: Applications of Statics

      When performing calculations, the first step is to determine if the system is, in fact, in equilibrium. Recall from the previous section that two conditions must be met for a system to be in equilibrium: the system must not be accelerating and the torque must be zero. The second step is to draw a free-body diagram of the system. It is important to determine all of the forces acting upon the system. The third step is to solve the problem by applying the relevant conditions of equilibrium: force is zero, and torque is zero.

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        Torques on a Seesaw Page
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        As you read, pay attention to Example 9.1, which shows how to do a statics problem. Here, children are balanced on a seesaw. We are given information about the masses of both children, and how far from the pivot point one child is sitting. We are asked to determine where the second child is sitting to balance.

        In Figure 9.8 we see that the children are balanced and therefore are at equilibrium. The free-body diagram shows that there is no net force, and no net rotational acceleration. To determine the distance of the second child from the pivot point, we use the torque equation, and set torque equal to zero. To determine the upward balancing force from the pivot point, we use the fact that net force equals zero to solve for the individual force at the pivot point.

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        Applications of Statics Page
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        As you read, notice that Example 9.2 shows a similar worked example of a statics problem. Here, a pole vaulter holds a pole at one end and we are asked to calculate the forces from each of the pole vaulter's hands. We take the same approach as in Example 9.1.

    • Unit 6 Assessment

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        Unit 6 Assessment Quiz
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        Take this assessment to see how well you understood this unit.

        • This assessment does not count towards your grade. It is just for practice!
        • You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
        • You can take this assessment as many times as you want, whenever you want.
      • quiz icon
        Unit 6 Assessment Quiz
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        Receive a grade

        Take this assessment to see how well you understood this unit.

        • This assessment does not count towards your grade. It is just for practice!
        • You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
        • You can take this assessment as many times as you want, whenever you want.

  • Unit 7: Work and Energy

    Energy describes the capacity of a physical system to perform work. It plays an essential role in everyday events and scientific phenomena. You can probably name many forms of energy: from the energy our food provides us, to the energy that runs our cars, to the sunlight that warms us on the beach. Not only does energy have many interesting forms, but it is involved in almost all phenomena and is one of the most important concepts of physics.

    Energy can change forms, but it cannot appear from nothing or disappear without a trace. Thus, energy is one of a handful of physical quantities that we say is conserved.

    Completing this unit should take you approximately 5 hours.

    • Upon successful completion of this unit, the student will be able to:

      • calculate the work done on an object by a force;
      • use the relationship between work done and the change in kinetic energy to make calculations;
      • state the work-energy theorem;
      • describe the concept of potential energy and how it relates to work;
      • compare and contrast conservative and non-conservative forces;
      • apply energy concepts to rotational motion;
      • solve dynamic linear and rotational problems using conservation of energy;
      • illustrate what physicists mean by power.

    • 7.1: Calculating Work and Force

      Work is done on a system when a constant applied force causes the system to be displaced or moved in the direction of the applied force. We can describe work using the equation W = Fd\cos\theta , where F is force, d is displacement, and \theta (the Greek letter theta) is the angle between F and d .

      From the equation for work, we can see that the unit for work must be the Newton-meter: the unit for force is the Newton and the unit for displacement (distance) is the meter. We define the Newton-meter as the unit joule. Consequently, we use joules as the unit for work and energy.
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        Introduction to Work Page
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        This video defines and explains the uses of work in the context of physics applications.

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        Work Example Problems Page
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        This video demonstrates how to solve work equations and some of their limitations.

    • 7.2: Work, Potential Energy, and Linear Kinetic Energy

      We define kinetic energy as the energy associated with motion. We calculate kinetic energy as {KE} = \frac{1}{2}mv^2 .

      When work is done on a system, energy is transferred to the system. We define net work as the total of all work done on a system by all external forces. We can think of the sum of all the external forces acting on a system as a net force, or F_{\mathrm{net}} .

      We can write the equation for net work in a similar way to how we wrote the equation for work earlier: W_{\mathrm{net}} = F_{\mathrm{net}}d\cos\theta , where W_{\mathrm{net}} is net work, F_{\mathrm{net}} is net force, d is displacement, and \theta is the angle between force and displacement.
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        Kinetic Energy and the Work-Energy Theorem Page
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        As you read, pay attention to the example of the forces on a box going across a conveyor belt in Figure 7.4. In this figure, we see different forces acting on the box. First, gravitational force is always present, which affects the weight ( w ) of the box. The normal force ( N ) balances the weight of the box. There is the applied force of the moving conveyor belt going to the right. Lastly, there is a horizontal frictional force from the rollers on the conveyor belt going back to the left. The weight and normal force cancel out. Therefore, the net force is the applied force minus the frictional force.

        See a worked example of calculating the kinetic energy for this box on a conveyor belt in Example 7.3. Work and kinetic energy are related in that work is the change in kinetic energy of an object. This is called the work-energy theorem. The work-energy theorem states that the net work on a system is the change of \frac{1}{2}mv^2 . That is: W = \frac{1}{2}mv^2_f - \frac{1}{2}mv^2_i , where m is mass, v_f is final velocity, and v_i is initial velocity. See a worked example in which the net force is calculated and used to determine the net work for the same system of the box on the conveyor belt in Example 7.4.
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        Work, Kinetic Energy, and Potential Energy Page
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        Watch this video, which accompanies what you just read. Greg Clements introduces the chapter and discusses how to calculate work.

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        More on the Work-Energy Theorem Page
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        Watch this video for a demonstration on how to use the work equation. Jennifer Cash also introduces how work relates to the change in kinetic energy.

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        More on Work and Energy Page
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        Watch this video for another take on the work equation.

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        Work as the Transfer of Energy Page
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        Watch this video for more on how work is a transfer, or kinetic energy.

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        Example of Work and Energy Page
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        This video presents an example of how to use the work-energy equation to solve problems.

    • 7.3: Conservative Forces and Potential Energy

      A non-conservative force is a force that depends on the path an object takes. In other words, a non-conservative force depends on how an object got from its initial state to its final state. Non-conservative forces change the amount of mechanical energy in a system. This differs from conservative forces, which do not depend on the path taken from initial to final state, and do not change the amount of mechanical energy in a system.

      An important example of a non-conservative force is friction. We know that friction is the force between two surfaces. We see friction when rolling a ball on a carpet versus a hardwood floor. The ball rolls farther on the hardwood floor than it does on a carpet. This is because the fuzzy carpet has more friction than the smooth hardwood. Friction converts some of the kinetic energy of the ball to thermal energy, or heat. As kinetic energy is converted to thermal energy, the balls slows to a stop.

      On the other hand, a conservative force is a force which does work that only depends on the beginning point and the end point of the system. The work done by a conservative force does not depend on the path the system takes to get from beginning to end. Conservative forces exist in ideal systems with no friction. An idealized spring that does not experience friction would be an example of conservative forces.

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        As you read, pay attention to Figure 7.15 for a comparison of conservative and non-conservative forces. In Figure 7.15 (a), a rock is being "bounced" on an ideal spring with no friction. The mechanical energy does not change, and the rock will continue bouncing indefinitely. In Figure 7.15 (b), the rock is thrown and lands on the ground. When it hits the ground, its kinetic energy is converted to thermal energy and sound. The rock can not "bounce" back up because its mechanical energy is not conserved.

        Gravity is a good example of a conservative force we use a lot in physics. Gravitational force is a conservative force because the work gravity does on an object does not depend on the path the object takes. Consequently, gravity is a good candidate to add into the work-energy theorem, where work is only done by gravity: W=Fd=mad

        Since the acceleration due to gravity is simply g and the direction of motion due to gravity is in the y-axis, we can further build the equation that represents work due to gravity: W=mg(\Delta y)=\Delta(mgy)

        Previously, we have discovered that work is also equal to the change in kinetic energy (see Section 7.2). So, we can now combine our equation for work due to gravity and our equation for work with respect to the change in kinetic energy: \Delta(mgy)=\Delta(\frac{1}{2})mv^{2} . The mgy in the equation is called the gravitational potential energy. We define potential energy as stored energy due to a system's position: PE=mgy .
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        Conservative Forces and Potential Energy Page
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        An example of an object with high potential energy is a compressed or stretched spring. When you let go of the compressed or stretched spring, the spring will release its potential energy as kinetic energy and go back to its usual shape. To calculate the potential energy of a spring, PE_{s} , we use the equation PE_{s} = \frac{1}{2}kx^{2} , where k is the spring constant and x is displacement from the spring's equilibrium. Read this text to see an example of a spring being stretched in Figure 7.10. The figure shows the work and potential energy associated with this.

        Mechanical energy is the sum of potential energy and kinetic energy of a system. Conservation of Mechanical Energy states that the sum of potential energy ( PE ) and kinetic energy ( KE ) is constant for a given system if only conservative forces act upon the system. We can write this in two different forms: KE + PE = \mathrm{Constant} or KE_{i} + PE_{i} = KE_{f} + PE_{f} . The second version of the equation can be more useful in describing changes from initial conditions ( KE_{i} and PE_{i} ) to final conditions ( KE_{f} and PE_{f} ). See the derivation of the conservation of mechanical energy from the work-energy theorem in equations 7.43, 7.44, 7.45, 7.46, 7.47, and 7.48.

        See a worked example of using conservation of mechanical energy to determine an object's speed in Example 7.8. In this example, we use the conservation of mechanical energy and the definitions of potential and kinetic energy to determine velocity. In these types of problems, it can be helpful to make a list of the information given in the problem to help determine what variable you can solve for.
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        Conservative Forces Page
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        Watch this video to learn more about what constitutes a conservative force.

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        More on Non-Conservative Forces Page
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        Watch this video to learn more about what constitutes a non-conservative force.

    • 7.4: Conservation of Energy

      The Law of Conservation of Energy states that the total energy in any process is constant. Energy can be transformed between different forms, and energy can be transferred between objects. However, energy cannot be created or destroyed. This is a broader law than the conservation of mechanical energy because this applies to all energy, not just energy when only conservative forces are applied.

      We can write the Law of Conservation of Energy as KE + PE + OE = \mathrm{Constant} or as KE_{i} + PE_{i} + OE_{i} = KE_{f} + PE_{f} + OE_{f}

      In the second equation, the KE_{i} , PE_{i} , and OE_{i} are initial conditions and KE_{f} , PE_{f} , and OE_{f} are final conditions. The new term, OE , is other energy. This is a collected term for all forms of energy that are not kinetic energy or potential energy. Other forms of energy include: thermal energy (heat), nuclear energy (used in nuclear power plants), electrical energy (used to power electronics), radiant energy (light), and chemical energy (energy from chemical reactions).

      When solving Conservation of Energy problems, it is important to identify the system of interest, and all forms of energy that can occur in the system. To do this, we need to first identify all forces acting on the system. Then, we can plug equations for different types of energy into the Law of Conservation of Energy equation to solve for the unknown in the problem.
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        Conservation of Energy Page
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        As you read, pay attention to the section Problem Solving Strategies for Energy for a step-by-step guide for solving these types of problems.

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        The Equation for Conservation of Mechanical Energy Page
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        Watch this video to learn about the conservation of energy equation in a lecture presentation format.

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        More on Conservation of Energy Page
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        Watch this video as it demonstrates solving for the conservation of energy equation for an object transitioning from gravitational potential energy to kinetic energy.

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        The Equation for Non-Conservative Work Page
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        Often, non-conservative forces come into play when dealing with motion. In this case, the conservation of mechanical energy does not hold. Watch the following video and read the following text to learn about how we can modify the conservation of energy equation to account for non-conservative forces.

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        Thermal Energy from Friction Page
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        These next two videos demonstrate a typical non-conservative force, friction, as it's used in solving the conservation of energy equation.

    • 7.5: Rotational Kinetic Energy

      Why do tornadoes spin so rapidly? The answer is that the air masses that produce tornadoes are themselves rotating, and when the radii of the air masses decrease, their rate of rotation increases. An ice skater increases their spin in an exactly analogous way. The skater starts their rotation with outstretched limbs and increases their spin by pulling them in toward their body. The same physics describes the spin of a skater and the wrenching force of a tornado. Clearly, force, energy, and power are associated with rotational motion.

      We cover these and other aspects of rotational motion in this unit. We will see that important aspects of rotational motion have already been defined for linear motion or have exact analogs in linear motion.

      We can write an equation for the rotational kinetic energy (the energy of rotational motion) as: KE_{\mathrm{rot}}=\frac{1}{2}I \omega^{2}

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        Rotational Kinetic Energy Book
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        As you read, pay attention to the diagram of a spinning disk in Figure 10.15. For the disk to spin, work must be done on the disk. The force acting upon the disk must be perpendicular to the radius of the disk, which we know is torque. We also know torque is related to moment of inertia. We can relate the work done on the disk to moment of inertia using the equation W=\tau\theta=I\alpha\theta .

        Example 10.8 shows how to calculate the net work for a rotating disk using this work equation. In the second part of the example, the rotational velocity is determined using the equation for rotational acceleration and moment of inertia. Lastly, it uses this equation to calculate the rotational kinetic energy.
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        Deriving Rotational Kinetic Energy Page
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        Watch this video to learn about the fundamental derivation of rotational kinetic energy and how it relates to linear kinetic energy.

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        This video discusses the formulation of rotational kinetic energy and its relation to linear kinetic energy from a more mathematical point of view, and gives an example of how to use the rotational kinetic energy equation.

    • 7.6: Power

      We define power as the rate at which work is done. We can write this as P = \frac{W}{\Delta t} , where w is work and \Delta t is the duration of the work being done. The unit for power is the watt, W. One watt equals one joule per second.

      Higher power means more work is done in a shorter time. This also means that more energy is given off in a shorter time. For example, a 60 W light bulb uses 60 J of work in a second, and also gives off 60 J of radiant and heat energy every second.
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        Power Page
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        We define power as the rate at which work is done. We can write this as P = \frac{W}{\Delta t} , where w is work and \Delta t is the duration of the work being done. The unit for power is the watt, W. One watt equals one joule per second.

        Higher power means more work is done in a shorter time. This also means that more energy is given off in a shorter time. For example, a 60 W light bulb uses 60 J of work in a second, and also gives off 60 J of radiant and heat energy every second.
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        More on Power Page
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        This video provides a brief introduction to the concept of power.

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        Work, Energy, and Power in Humans Book
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        Read these texts to learn how energy is transferred and transformed in humans and in society.

    • Unit 7 Assessment

      • quiz icon
        Unit 7 Assessment Quiz
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        Take this assessment to see how well you understood this unit.

        • This assessment does not count towards your grade. It is just for practice!
        • You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
        • You can take this assessment as many times as you want, whenever you want.

  • Unit 8: Momentum and Collisions

    We use the term momentum in various ways in everyday language. For example, we often speak of sports teams gaining and maintaining the momentum to win. Generally, momentum implies a tendency to continue on course (to move in the same direction) and is associated with mass and velocity. Momentum has its most important application when analyzing collision problems. Like energy, it is important because it is conserved. Only a few physical quantities are conserved in nature, and studying them yields fundamental insight into how nature works, as we shall see during our study of momentum.

    Completing this unit should take you approximately 4 hours.

    • Upon successful completion of this unit, the student will be able to:

      • define linear momentum;
      • explain how momentum and force are related;
      • use Newton's Second Law of Motion in terms of momentum;
      • describe the relationship between impulse and momentum;
      • define elastic, inelastic, and totally inelastic collisions;
      • use conservation of momentum to solve linear collision problems;
      • use conservation of momentum to solve situations involving rotation; and
      • give several everyday examples of conservation of angular momentum.

    • 8.1: Linear Momentum

      We define linear momentum as the product of an object's mass and velocity. It can be written as p = mv , where p is linear momentum, m is mass, and v is velocity.

      Linear momentum is a vector quantity because velocity is a vector quantity, and the linear momentum will have the same direction as the velocity. The units for linear momentum are kgm/s.
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        Linear Momentum and Force Page
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        This text goes into the concepts of linear momentum, impulse, and how force is used to change momentum over time.

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        Momentum, Impulse, and the Conservation of mV Page
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        This video discusses some of the concepts we will explore later in this unit.

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        More on Momentum Page
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        This video gives another way of thinking about momentum.

    • 8.2: Momentum and Newton's Second Law

      In Unit 4 we learned to write Newton's Second Law of Motion as F= ma . While this is the most common way to write and use this law, it was not how Newton originally wrote it. Newton wrote this law in terms of momentum rather than force and acceleration: F_{net} = \frac{\Delta p}{\Delta t}

      This shows that the net force equals the change in momentum divided by the change in time. This equation certainly appears different from the familiar F=ma we used in Unit 4.
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        Momentum and Newton's Second Law Page
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        As you read, pay attention to how we can derive F=ma from the Second Law in terms of momentum in equations 8.9, 8.10, 8.11, and 8.12. Furthermore, we define impulse as change in momentum. Using Newton's Second Law of Motion, we can write this as \Delta p=F_{net,avg} \Delta t .

        See how to use Newton's Second Law in terms of momentum in Example 8.2. This problem calculates the force applied to a tennis ball: there is a change of velocity of the ball but no change in mass, so pay special attention to how change in momentum is calculated in equation 8.14.

        When we calculate impulse, we assume the net force is constant during the time we are interested in. In reality, force is rarely constant. For example, in Example 8.2, we assumed the force on the tennis ball was constant over time. In reality, the force on the tennis ball probably changed from the beginning to the end of the swing of the tennis racquet. Nevertheless, the change in force was probably not significant, and we assume it is constant to make our calculations easier.
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        Impulse Page
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        Read this text as it expands on linear momentum and Newton's Second Law to define a new quantity, impulse.

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        More on Impulse Page
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        Also This video introduces the concept of impulse.

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        Example of Impulse and Momentum in Dodgeball Page
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        This video describes how to use the impulse-force equation to solve problems.

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        Force vs. Time Graphs Page
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        This video presents a graphical analysis of force and time and how they relate to impulse.

    • 8.3: Elastic, Inelastic, and Totally Inelastic Collisions

      When two or more objects interact physically, we say the objects collide or experience a collision. Here, we consider three types of collisions for solving physics problems. They are all based on the energy transfer in the collisions. By definition, an elastic collision is a collision where the internal kinetic energy is conserved in the interaction. So, in an elastic collision, all the kinetic energy remains kinetic energy. That is, no kinetic energy is converted to heat, friction, or other types of energy.

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        Elastic Collisions in One Dimension Page
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        As you read, pay attention to the diagram of two metal boxes interacting in an elastic collision on an ice surface in Figure 8.6.

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        Conserving Momentum in Elastic Collisions Page
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        Watch this video, which accompanies what you just read.

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        What are Inelastic and Elastic Collisions? Page
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        Unlike an elastic collision, an inelastic collision is a collision where the internal kinetic energy is not conserved. In inelastic collisions, some kinetic energy of the colliding objects is lost to friction, heat, or even work. Inelastic collisions are what we mostly observe in the real world. Watch this video for an overview of inelastic and elastic collisions.

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        Inelastic Collisions in One Dimension Page
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        Read this text. As we learned in the previous video, in reality, no collisions are perfectly elastic because some kinetic energy is always "lost" by being converted to other forms of energy. Another example of an elastic collision is if two balls collide on a smooth icy surface. Because the ice has almost no friction, little kinetic energy would be lost to friction.

        See an example of two blocks experiencing a totally inelastic collision in Figure 8.8. See a good example of an inelastic collision in Figure 8.9. In this example, a hockey goalie stops a puck in the net. Although the ice surface is essentially frictionless, some kinetic energy of the puck is converted to heat and sound as the goalie stops it. A totally inelastic collision (also called a perfectly inelastic collision) is an inelastic collision where the objects "stick together" upon colliding.

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        Conserving Momentum in Inelastic Collisions in Two Dimensions Page
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        Watch this video, which accompanies what you just read.

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        Elastic and Inelastic Collisions Page
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        This video compares elastic and inelastic collisions.

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        Perfectly Inelastic Collisions Page
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        This video looks more closely at the case for perfectly inelastic collisions. What makes something perfectly inelastic is that the objects stick together after the collision. This means they have the same final velocity.

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        Elastic and Inelastic Collisions in One Dimension Page
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        Watch this video for another explanation of elastic and inelastic collisions in one dimension.

    • 8.4: Solving Problems Involving Conservation of Linear Momentum in Collisions

      When solving problems for elastic collisions, it is important to remember that the kinetic energy is conserved. Therefore, the total kinetic energy at the start of the collision must equal the total kinetic energy at the end of the collision. We can write this as \frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2 = \frac{1}{2}m_1 v_1^{'2} + \frac{1}{2}m_2 v_2^{'2}

      Moreover, we know that momentum must be conserved in the collision. Therefore, the total momentum at the start of the collision must equal the total momentum at the end of the collision. That is, for two objects (object one and two) colliding, we can write \frac{1}{2}m_1 v_1 + \frac{1}{2}m_2 v_2 = \frac{1}{2}m_1 v_1^{'} + \frac{1}{2}m_2 v_2^{'} . Using conservation of momentum, we can usually set up these problems so we only have to solve for one unknown.
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        Elastic Collisions in One Dimension Page
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        As you read, pay attention to Example 8.4. In this example, one of the objects is initially at rest (its velocity equals zero), so it does not have an initial momentum. This lets us simplify the conservation of energy momentum equations. Then, by using the equations for conservation of energy and momentum, we can solve for final velocity after collision.

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        Calculating Velocity Page
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        Read this text, which shows that we can also solve for final velocities after inelastic collisions. In these problems, it is important to remember that kinetic energy is not conserved but momentum is conserved.

        In Example 8.5(a), the conservation of momentum equation is used to determine the final velocity of the object (the hockey goalie) in an inelastic collision. In inelastic collisions, some kinetic energy is converted to other forms of energy. The energy difference before and after collision can be calculated to determine how much kinetic energy was lost.

        In Example 8.5(b), the amount of energy lost is calculated. The total kinetic energy in the system is calculated before and after collision based on the mass and velocities of the objects. The difference in kinetic energy shows how much kinetic energy was converted to other forms of energy during the collision. Example 8.6 is similar.

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        Linear Momentum and the Conservation of Momentum Page
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        This video gives an overall overview of linear momentum, impulse, and the conservation of momentum law.

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        Example: Bouncing Fruit Colliding Page
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        This video gives an example of using the conservation of momentum law to solve collision problems (problems that deal with two objects combining).

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        Example: An Ice Skater Throws a Ball Page
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        This video gives an example of using the conservation of momentum to solve explosion problems (problems that deal with two objects separating from each other).

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        Two-Dimensional Momentum Page
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        These videos expand on the conservation of momentum law for two-dimensional collision problems. Note that the conservation of momentum simply needs to be applied twice: once for each of the two dimensions of motion between objects.

    • 8.5: Conservation of Angular Momentum

      We define angular momentum as L=I\omega . It is similar to the momentum defined for linear motion. As such, angular momentum in a system is conserved in the same way that linear momentum is conserved. Therefore, we can say that L=L' , where L is the initial angular momentum in a system and L' is the final angular momentum in the system. We can also write this as:

      I\omega=I'\omega'  

      We see conservation of angular momentum in many everyday examples. As you read, pay attention to the example of the spinning figure skater in Figure 10.23. In the first picture, the figure skater is spinning with her arms out on a frictionless ice surface. In the second picture, she pulls her arms in, and her rotational velocity increases. When the figure skater pulls in her arms, she lowers her moment of inertia. Because angular momentum is conserved, because her moment of inertia decreases, her angular velocity must therefore increase.
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        Angular Momentum and Its Conservation – IP Book
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        Read this to understand angular momentum, how torque plays a role, and how angular momentum is conserved without torque.

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        Controlling Angular Velocity on a Rotating Stool Page
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        This video gives a demonstration and a brief explanation behind the conservation of angular momentum.

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        Conserving Angular Momentum Page
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        This video goes into solving for the conservation of angular momentum for a particular situation.

    • Unit 8 Assessment

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        Unit 8 Assessment Quiz
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        Take this assessment to see how well you understood this unit.

        • This assessment does not count towards your grade. It is just for practice!
        • You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
        • You can take this assessment as many times as you want, whenever you want.

  • Study Guide

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