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.
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 . 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.
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.
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.
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.
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 . 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 .
Read this text for examples of how to calculate physical quantities and units of measurement.
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.
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.
Read this text to learn more about uncertainty, accuracy, precision and significant figures.
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: . To put the entire scientific notation together, we write: .
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: . To put the entire scientific notation together, we write: .
We can also convert values written in scientific notation to decimal notation. Consider the number: . 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: .
We can also do this for small numbers written in scientific notation. Consider the example: . 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: .
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.
This video shows some practical applications to scientific notation. Pay close attention to the various ways of writing the same number.
Watch this video to review converting scientific notation to standard notation.
Take this assessment to see how well you understood this unit.