Topic Name Description
Course Syllabus Course Syllabus
1.1: Scientific Theory, Law, and Models An Introduction to Physics

Read this text, which introduces these concepts.

1.2: Physical Quantities and Units Physical Quantities and Units

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 Unit Conversion and Dimensional Analysis

Read this text for examples of how to calculate physical quantities and units of measurement.

Metric Units, Converting Units, Significant Figures

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 Accuracy, Precision, and Significant Figures

Read this text to learn more about uncertainty, accuracy, precision and significant figures.

1.5: Scientific Notation Review of Scientific Notation

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.

Applying Scientific Notation

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

Converting Scientific Notation to Standard Notation

Watch this video to review converting scientific notation to standard notation.

2.1: Vectors, Scalars, and Coordinate Systems Vectors, Scalars, and Coordinate Systems

Read this text to learn more about the difference between a scalar and vector quantity.

2.2: Instantaneous and Average Values for Physical Quantities Time, Velocity, and Speed

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 Displacement

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

Distance and Displacement as Scalar Vectors

This lecture accompanies what you just read.

Displacement vs. Distance

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

Displacement Time Graphs

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

Vectors and Scalars

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

2.4: Speed and Velocity Time, Velocity, and Speed

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.

More on Speed and Velocity

This lecture accompanies what you just read.

Speed and Velocity vs. Distance and Displacement

Watch this video to discover how average speed and velocity relate to distance and displacement.

2.5: Motion with Constant Acceleration Acceleration

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.

More on Acceleration

This lecture accompanies what you just read.

Motion Equations for Constant Acceleration in One Dimension

Read this text for more examples and practice on how to solve motion equations for constant acceleration.

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

Read this text for more about falling objects, gravity, and freefall. We will discuss how to calculate the examples here in the next section.

Vertical Motion in Free Fall

This lecture accompanies what you just read.

Zero Gravity Demonstration

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 Kinematic Equations for Objects in Free Fall

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.

Constant Acceleration Equations

Watch this video for details on how kinematic equations were derived.

More on Free Fall

Watch this video for details on kinematic equations and its relationship with freefall.

Problem-Solving Basics for One-Dimensional Kinematics

Read this text for a step-by-step guide on how to solve problems of constant acceleration using kinematic equations.

Kinematic Equations in Constant Acceleration

This lecture accompanies what you just read.

Displacement with Constant Acceleration
This video goes into more detail about constant acceleration.
2.8: Graphical Analysis Graphical Analysis of One-Dimensional Motion

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.

Interpreting Velocity Graphs
Watch this video for details regarding the velocity versus time graphs and how they relate to acceleration.
3.1: Introduction to Kinematics in Two Dimensions using Vectors Kinematics in Two Dimensions

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

Unit Vectors and Engineering Notation

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 Vector Addition and Subtraction

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.

Vector Addition Using the Graphical Method

This lecture accompanies what you just read.

3.3. Adding Vectors Analytically: Determining the Components, Magnitude, and Direction of a Vector Analytical Methods for Vector Addition and Subtraction

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.

More on Vector Addition

This lecture accompanies what you just read.

Vector Components on a Grid

This video discusses how vectors are represented as components in the x- and y-axes.

Projectiles at an Angle

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

Visualizing Vectors in Two Dimensions

This video describes how to convert vectors from magnitude-direction form into component form, and vice versa.

Unit Vector Notation

This video demonstrates how to add vectors using the graphical and analytical methods.

3.4: Projectile Motion and Trajectory Projectile Motion

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

More on Projectile Motion

This lecture accompanies what you just read.

Another Way to Determine Time In Air

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.

Horizontally-Launched Projectiles

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

Launching and Landing at Different Elevations

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.

Total Displacement for a Projectile

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

Total Final Velocity for a Projectile

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.

Projectiles on an Incline

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

4.1: Newton's First Law of Motion Newton's First Law of Motion and Inertia

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.

Force and Newton's Laws

This lecture accompanies what you just read.

The Historical Context

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

More on Newton's First Law of Motion

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

4.2: Newton's Second Law of Motion Newton's Second Law of Motion

This video reviews how Newton's Second Law was derived using concepts that are familiar from previous units.

Using Newton's Second Law

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.

F=ma

This lecture accompanies what you just read.

Examples of Newton's Second Law

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.

More on Newton's Second Law

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

4.3: Free-Body Diagrams The Concept of Force

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 Symmetry in Forces

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.

Newton's Third Law and F=ma

This lecture accompanies what you just read.

More on Newton's Third Law

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

Examples of Newton's Third Law

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 Problem-Solving Strategies

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.

Further Applications of Newton's Laws of Motion

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.

Identifying and Labeling Types of Forces

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

Mass and Weight

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.

Characteristics of Forces

This video demonstrates the characteristics of the forces of tension, friction, weight, and normal when solving problems related to forces.

Gravity and Weight

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.

Resultant Forces
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 Newton's Universal Law of Gravitation

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.

Introduction to Gravity

This video introduces the concept of the Universal Law of Gravitation and relates it to its local law version ($F=mg$).

Introduction to Newton's Law of Gravitation

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.

Gravity for Astronauts in Orbit

Watch this video for detailed analysis of gravity as it applies to astronauts flying high above the atmosphere.

Would a Brick or Feather Fall Faster?

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 Normal Force

Read this section, which discusses normal force.

Identifying and Labeling Types of Forces

Watch this video until 4:37.

More on Normal Force

This video explains normal force: the force of contact. Again, "normal" essentially means perpendicular. The opposing force is the normal force.

Normal Force and Contact Force

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

Normal Force in an Elevator

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.

Multiple Forces Alongside Normal Force

These two videos describe more complicated situations dealing with multiple forces (some going diagonal) accompanying normal force.

Ice Accelerating Down An Incline

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

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.

Identifying and Labeling Types of Forces

Watch this video from 6:34 to 7:03.

Tension Forces

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 Identifying and Labeling Types of Forces

Watch this video from 4:37 to 6:34.

Friction

The text explains the fundamentals of friction that we discussed earlier, but in more detail.

Kinetic and Static Friction Forces

This video demonstrates the difference between kinetic and static friction.

Friction and Force from Springs

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.

Comparing Static and Kinetic Friction

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

Examples of Static and Kinetic Friction

This video explains how to solve basic friction problems involving static and kinetic cases.

5.1: Centripetal Force Centripetal Acceleration

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.

Circular Motion and Centripetal Acceleration

This lecture accompanies what you just read.

Centripetal Force
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.
Tennis Ball on a String
Watch this video for a demonstration of centripetal force.
More on Centripetal Force

This lecture accompanies what you just read.

Visualizing Centripetal Acceleration

Watch this video to see how the equation for centripetal force is derived.

More on Centripetal Force And Acceleration

Watch this video as it goes into the concept of centripetal acceleration and centripetal force.

Example: Loop de Loop

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 Satellites and Kepler's Laws

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.

Kepler's Three Laws of Planetary Motion
This video accompanies the text on Kepler's Three Laws.
5.3: Angular Position, Velocity, and Acceleration Rotation Angle and Angular Velocity

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.

Angular Acceleration

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 Kinematics of Rotational Motion

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.

6.1: Conditions for Equilibrium The First Condition for Equilibrium

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.

Static Equilibrium, Torque, and Stability

Watch this video, which accompanies what you just read.

6.2: Torque Rotational Inertia

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.

Rotational Kinematics and Dynamics

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.

The Second Condition for Equilibrium

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.

Moment of Inertia
This video discusses the concept of moment of inertia.
The Race Between a Ring and a Disc
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 Torques on a Seesaw

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.

Applications of Statics

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.

7.1: Calculating Work and Force Introduction to Work

This video defines and explains the uses of work in the context of physics applications.

Work Example Problems

This video demonstrates how to solve work equations and some of their limitations.

7.2: Work, Potential Energy, and Linear Kinetic Energy Kinetic Energy and the Work-Energy Theorem

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.

Work, Kinetic Energy, and Potential Energy

Watch this video, which accompanies what you just read. Greg Clements introduces the chapter and discusses how to calculate work.

More on the Work-Energy Theorem

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.

More on Work and Energy

Watch this video for another take on the work equation.

Work as the Transfer of Energy

Watch this video for more on how work is a transfer, or kinetic energy.

Example of Work and Energy

This video presents an example of how to use the work-energy equation to solve problems.

7.3: Conservative Forces and Potential Energy Non-Conservative Forces

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: $PE=mgy$.

Conservative Forces and Potential Energy

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.

Conservative Forces

Watch this video to learn more about what constitutes a conservative force.

More on Non-Conservative Forces

Watch this video to learn more about what constitutes a non-conservative force.

7.4: Conservation of Energy Conservation of Energy

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.

The Equation for Conservation of Mechanical Energy

Watch this video to learn about the conservation of energy equation in a lecture presentation format.

More on Conservation of Energy

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

The Equation for Non-Conservative Work

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.

Thermal Energy from Friction

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 Rotational Kinetic Energy

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.

Deriving Rotational Kinetic Energy

Watch this video to learn about the fundamental derivation of rotational kinetic energy and how it relates to linear kinetic energy.

More on Rotational Kinetic Energy

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 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.

More on Power

This video provides a brief introduction to the concept of power.

Work, Energy, and Power in Humans

Read these texts to learn how energy is transferred and transformed in humans and in society.

8.1: Linear Momentum Linear Momentum and Force

This text goes into the concepts of linear momentum, impulse, and how force is used to change momentum over time.

Momentum, Impulse, and the Conservation of mV

This video discusses some of the concepts we will explore later in this unit.

More on Momentum

This video gives another way of thinking about momentum.

8.2: Momentum and Newton's Second Law Momentum and Newton's Second Law

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.

Impulse

Read this text as it expands on linear momentum and Newton's Second Law to define a new quantity, impulse.

More on Impulse

Also This video introduces the concept of impulse.

Example of Impulse and Momentum in Dodgeball

This video describes how to use the impulse-force equation to solve problems.

Force vs. Time Graphs

This video presents a graphical analysis of force and time and how they relate to impulse.

8.3: Elastic, Inelastic, and Totally Inelastic Collisions Elastic Collisions in One Dimension

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.

Conserving Momentum in Elastic Collisions

Watch this video, which accompanies what you just read.

What are Inelastic and Elastic Collisions?

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.

Inelastic Collisions in One Dimension

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.

Conserving Momentum in Inelastic Collisions in Two Dimensions

Watch this video, which accompanies what you just read.

Elastic and Inelastic Collisions

This video compares elastic and inelastic collisions.

Perfectly Inelastic Collisions

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.

Elastic and Inelastic Collisions in One Dimension

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 Elastic Collisions in One Dimension

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.

Calculating Velocity

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.

Linear Momentum and the Conservation of Momentum

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

Example: Bouncing Fruit Colliding

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

Example: An Ice Skater Throws a Ball

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).

Two-Dimensional Momentum

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 Angular Momentum and Its Conservation – IP

Read this to understand angular momentum, how torque plays a role, and how angular momentum is conserved without torque.

Controlling Angular Velocity on a Rotating Stool

This video gives a demonstration and a brief explanation behind the conservation of angular momentum.

Conserving Angular Momentum

This video goes into solving for the conservation of angular momentum for a particular situation.

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