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All measured quantities carry some uncertainty in their values. When working with the values of quantities it is important to keep proper account of the digits that are reliably known. Such digits are called significant figures. The rules for working with significant figures are as follows: * Multiplication and Division: The number of significant figures in the result of a multiplication or division equals the number of significant figures in the factor containing the fewest significant figures.
Example 1-3 Driving in a Residential Zone: On most residential streets in the United States the speed limit is 25 mi/h (= 11 m/s). If a car drives down a neighborhood side street at the legal speed limit for 120.46 s, how much distance does the car cover? Picture the Problem Our sketch shows the car moving along a straight road.  Strategy The distance traveled is the speed multiplied by the time of travel. Solution
Insight There are two important things to notice about the result. First, despite the fact that the time is known to five significant figures, the speed is known only to two and so the result has only two significant figures. Second, the final two zeros in the value 1300 are not significant. They must be written, however, to have the proper magnitude of the value. It can often be unclear whether or not such zeros are significant. This problem can be avoided by using scientific notation (discussed next). Example 1-4 Significant Figures A calculation involves the addition of two measured distances d1 = 1250 m and d2 = 336 m. If each measurement is given to three significant figures, what is the result of the calculation? Solution Adding the two distances we get d1 + d2 = 1250 m = 336 m = 1590 m. The answer is not 1586 because even though 6 is significant in 336 m, the one's place of 1250 m (the 0) is not significant, so the one's place of the result cannot be significant. You may wonder about the fact that there is no significant figure in the thousand's place of 336 m because the value requires no digit there; however, because there is no digit there we know that place with certainty. Round-off Error Be aware that to avoid excessive round-off error you should round only to the proper number of significant figures at the very end of a calculation. In Example 1-4, if the distance calculated is only an intermediate step in a longer calculation, then the value 1586 m should be used in the subsequent steps. In general, keep at least one additional digit for values calculated in intermediate steps. Another, even better, approach is not to calculate intermediate values numerically but just carry through the formulas inserting numerical values only at the end. Example 1–5 Don’t Round-off too Soon A cardboard box has measurements of L = 1.92 m, and W = 0.725 m. Its height is H = 1.88 m. (a) Calculate the area (A) of the base of the box. (b) Calculate its volume (V) using the result of (a). (c) Calculate its volume using the formula for volume. Picture the Problem The diagram shows a box representing the box whose base area and volume we wish to determine.  Strategy First calculate the area of the base. Solution
Insight The answers to parts (b) and (c) differ in the final digit. Which one is correct? Part (c) is correct because the full values were used. The round-off to three significant figures in part (a) is the reason for the difference. Scientific Notation A very useful way of writing numerical values is to use scientific notation.
In this notation a value is written as a number of order unity (meaning that
only one digit is left of the decimal point) times the appropriate power of
10. The value of scientific notation is that it allows for quick identification
of the order-of-magnitude (power of ten) of a quantity, calculations
are often easier to perform when the values are listed this way, and it removes
any ambiguity in the number of significant figures. For example, if the number
3500 has only one significant figure, we write it is as 4 x 103;
if it has two, we write 3.5 x 103; if it has three we write 3.50
x 103; and if it has four significant figures we write 3.500 x 103. Exercise 1–6 Scientific Notation Write the following quantities using scientfic notation assuming three significant figures. (a) 0.00250 m, (b) 12,060 m, (c) 451 m, (d) 8.00 m, (e) .00003593 m Answer: (a) 2.50 x 10-3 m (b) 1.21 x 104 m (c) 4.51 x 102 m (d) 8.00 x 100 m (e) 3.59 x 10-5 m When the value is already of order unity, as with part (d), the power of ten
is often dropped. In such cases, the fact that the two zeros are written after
the decimal point suggests that they are significant figures. Practice Quiz
1–5 Converting Units Even though we predominantly use SI units, it will often be necessary to convert between SI and other units. A conversion can be accomplished using a conversion factor that is constructed by knowing how much of a quantity in one unit equals that same quantity in another unit. A conversion factor is a ratio of equal quantities written such that, when multiplied by a quantity, the undesired unit algebraically cancels leaving only the desired unit. This concept is best illustrated by example. Example 1–7 Volume of a Box A typical cardboard box provided by moving companies measures 1.50 ft x 1.50 ft x 1.33 ft. Determine the volume (V) of clothes that you can pack into this box in cubic meters. Picture the Problem The diagram represents the box whose volume we wish to determine.
 Strategy We first calculate the volume in the given units, determine the conversion factor, then convert the volume to cubic meters. Solution
Insight In the final step, the unit ft3 cancels just as numbers would. Setting up this cancellation is the crucial step in unit conversion. You will get plenty of practice converting units in your study of physics.
Practice Quiz
1-7 Problem Solving in Physics Solving physics problems is a logical and creative endeavor for which there is no set prescription; however, there are certain practices that help this creative process to flourish. First, a careful reading of the problem is necessary to fully grasp the question being posed and the information being given. It is often useful to separately write out all the given and required information; several of the solved examples in this study guide illustrate that approach. It is also good practice to make a sketch of the problem and visualize the physics taking place. A correct mental picture of the problem takes you a long way toward a correct solution. Next, map out your strategy for the solution. Here, you basically solve the problem logically before doing it mathematically. For the mathematical solution you need to identify and solve the appropriate equations for the relevant physics. Finally, you should check and explore your result to be sure that the answer makes sense in the context of the problem.
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Introduction to Physlet Illustrations