## PHYS101 Study Guides

### 1a. Explain the difference between a theory and a law

• Define a scientific law, scientific theory, and scientific model.
• Give an example of a scientific law and a scientific theory.
• What are some similarities between scientific laws and theories? What is the main difference between them?

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.

Review the section Models, Theories, and Laws; The Role of Experimentation.

### 1b. Perform unit conversions using both metric and traditional U.S. units

• Define physical quantity. What makes something a physical quantity?
• Define SI fundamental units. How are they related to SI derived units?
• List the metric prefixes and their order of magnitude for commonly used units.
• Perform unit conversions between different metric units using metric prefixes and orders of magnitude.
• Perform unit conversions between metric units and U.S. 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).

SI units are standardized units of measure for different measured quantities. The fundamental SI units are the following:

 Mass Kilogram (kg) Length Meter (m) Time Second (s) Electric current Ampere (A)

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 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.

Review an extensive list of the metric prefixes in Table 1.2 Metric Prefixes for Powers of 10 and their Symbols.

Using the metric system to convert between units requires a knowledge of the prefixes and their order of magnitude. For example, consider length. The base unit for length is the meter (m). In Table 1.2, we see that one centimeter (cm) is 10-2 m.

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). To determine how to write this equivalence as a fraction, we need to determine what should be the numerator and what should be the denominator. 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 fraction with the unit you want to end up with in the numerator, and the unit you were given in the denominator. This will result in the answer being in the unit you want:

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

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 in the following way:

$\mathrm{5.0\ m\times\frac{1\ cm}{10^{-2}\ m}=500\ cm}$

The meter unit cancels out in the above 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 for 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 as above:

$\mathrm{5.5\ feet\times\frac{12\ inches}{1\ foot}=66\ inches}$

### 1c. Write numbers expressed in decimals as scientific notation and write numbers expressed in scientific notation as decimals

• Convert numbers to scientific notation. What do the different parts of scientific notation mean?
• Convert numbers written in scientific notation to decimal form.
• Perform calculations with numbers written in 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: 105.

To put the entire scientific notation together, we write: 1.25 × 105 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 × 10-6 m.

We can also convert values written in scientific notation to decimal notation. Consider the number: 5.0 × 103 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 m.

We can also do this for small numbers written in scientific notation. Consider the example: 4.2 × 10-4 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 m.

### 1d. Solve problems with the values of the most common metric prefixes

• Perform unit conversions between different metric units using metric prefixes and orders of magnitude.

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 (see Table 1.2 Metric Prefixes for Powers of 10 and their Symbols). To determine how to write this equivalence as a fraction, we need to determine what should be the numerator and what should be the denominator. That is, we could write 1 cm/10-2 m or we could write 10-2 m/1 cm.

We 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 fraction with the unit you want to end up with in the numerator and the unit you were given in the denominator. This will result in the answer being in the unit you want:

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

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 in the following way:

$\mathrm{5.0\ m\times\frac{1\ cm}{10^{-2}\ m}=500\ cm}$

The meter unit cancels out in the above calculation. Because the meter unit cancels out, we are left with cm as the unit of the answer.

For another example, we can determine the number of seconds in 424 nanoseconds (ns). In Table 1.2 we see that 1 ns = 10-9 s. Our given is in ns, so we need to write our unit conversion factor with ns in the denominator to cancel the unit out. We perform the calculation the following way:

$\mathrm{424\ ns\times\frac{10^{-9}\ s}{1\ ns}=4.24\times 10^{-7}\ s}$

### Unit 1 Vocabulary

• Derived SI units
• Dimensional analysis
• Fundamental SI units
• Metric prefix
• Metric system
• Order of magnitude
• Physical quantity
• Scientific law
• Scientific model
• Scientific notation
• Scientific theory
• SI fundamental units
• Unit conversion factor
• Units