Equations, Formulas, and Conversion Factors for Physics Help (page 2)
Some Simple Equations
Here are some simple equations containing only numbers. Note that these are true no matter what.
3 = 3
3 + 5 = 4 + 4
1,000,000 = 10 6
− (−20) = 20
Once in a while you’ll see equations containing more than one equals sign and three or more parts. Examples are
3 + 5 = 4 + 4 = 10 − 2
1,000,000 = 1,000 × 1,000 = 10 3 × 10 3 = 10 6
−(−20) = −1 × (−20) = 20
All the foregoing equations are obviously true; you can check them easily enough. Some equations, however, contain variables as well as numbers. These equations are true only when the variables have certain values; sometimes such equations can never be true no matter what values the variables attain. Here are some equations that contain variables:
x + 5 = 8
x = 2 y + 3
x + y + z = 0
x 4 = y 5
y = 3 x − 5
x 2 + 2 x + 1 = 0
Variables usually are represented by italicized lowercase letters from near the end of the alphabet.
Constants can be mistaken for variables unless there is supporting text indicating what the symbol stands for and specifying the units involved. Letters from the first half of the alphabet often represent constants. A common example is c , which stands for the speed of light in free space (approximately 299,792 if expressed in kilometers per second and 299,792,000 if expressed in meters per second). Another example is e , the exponential constant, whose value is approximately 2.71828.
Some Simple Formulas
In formulas, we almost always place the quantity to be determined all by itself, as a variable, on the left-hand side of an equals sign and some mathematical expression on the right-hand side. When denoting a formula, it is important that every constant and variable be defined so that the reader knows what the formula is used for and what all the quantities represent.
One of the simplest and most well-known formulas is the formula for finding the area of a rectangle (Fig. 1-1). Let b represent the length (in meters) of the base of a rectangle, and let h represent the height (in meters) measured perpendicular to the base. Then the area A (in square meters) of the rectangle is
A = bh
A similar formula lets us calculate the area of a triangle (Fig. 1-2). Let b represent the length (in meters) of the base of a triangle, and let h represent the height (in meters) measured perpendicular to the base. Then the area A (in square meters) of the triangle is
A = bh /2
Consider another formula involving distance traveled as a function of time and speed. Suppose that a car travels at a constant speed s (in meters per second) down a straight highway (Fig. 1-3). Let t be a specified length of time (in seconds). Then the distance d (in meters) that the car travels in that length of time is given by
d = st
If you’re astute, you will notice something that all three of the preceding formulas have in common: All the units “agree” with each other. Distances are always given in meters, time is given in seconds, and speed is given in meters per second. The preceding formulas for area will not work as shown if A is expressed in square inches and d is expressed in feet. However, the formulas can be converted so that they are valid for those units. This involves the insertion of constants known as conversion factors .
Refer again to Fig. 1-1. Suppose that you want to know the area A in square inches rather than in square meters. To derive this answer, you must know how many square inches comprise one square meter. There are about 1,550 square inches in one square meter. Thus we can restate the formula for Fig. 1-1 as follows: Let b represent the length (in meters) of the base of a rectangle, and let h represent the height (in meters) measured perpendicular to the base. Then the area A (in square inches) of the rectangle is
A = 1,550 bh
Look again at Fig. 1-2. Suppose that you want to know the area in square inches when the base length and the height are expressed in feet. There are exactly 144 square inches in one square foot, so we can restate the formula for Fig. 1-2 this way: Let b represent the length (in feet) of the base of a triangle, and let h represent the height (in feet) measured perpendicular to the base. Then the area A (in square inches) of the triangle is
A = 144 bh /2
= (144/2) bh
= 72 bh
Give Fig. 1-3 another look. Suppose that you want to know how far the car has traveled in miles when its speed is given in feet per second and the time is given in hours. To figure this out, you must know the relationship between miles per hour and feet per second. To convert feet per second approximately to miles per hour, it is necessary to multiply by 0.6818. Then the units will be consistent with each other: The distance will be in miles, the speed will be in miles per hour, and the time will be in hours. The formula for Fig. 1-3 can be rewritten: Suppose that a car travels at a constant speed s (in feet per second) down a straight highway (see Fig. 1-3). Let t be a certain length of time (in hours). Then the distance d (in miles) that the car travels in that length of time is given by
d = 0.6818 st
You can derive these conversion factors easily. All you need to know is the number of inches in a meter, the number of inches in a foot, the number of feet in a mile, and the number of seconds in an hour. As an exercise, you might want to go through the arithmetic for yourself. Maybe you’ll want to derive the factors to greater precision than is given here.
Conversion factors are not always straightforward. Fortunately, databases abound in which conversion factors of all kinds are listed in tabular form. You don’t have to memorize a lot of data. You can simply look up the conversion factors you need. The Internet is a great source of this kind of information. At the time of this writing, a comprehensive conversion database for physical units was available at the following location on the Web:
If you’ve used the Web very much, you know that uniform resource locators (URLs) are always changing. If the preceding URL does not guide you to conversion factors, point your browser to the National Institute of Standards and Technology (NIST) home page and search the site for tables of conversion factors:
Practice problems for these concepts can be found at: Equations, Formulas, And Vectors for Physics Practice Test
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