Physics For Scientists And Engineers 6E - part 4

1.6 Estimates and Order-of-Magnitude

Calculations

It is often useful to compute an approximate answer to a given physical problem even
when  little  information  is  available.  This  answer  can  then  be  used  to  determine
whether or not a more precise calculation is necessary. Such an approximation is usu-
ally  based  on  certain  assumptions,  which  must  be  modified  if  greater  precision  is
needed.  We  will  sometimes  refer  to  an order  of  magnitude of  a  certain  quantity  as  the
power  of  ten  of  the  number  that  describes  that  quantity.  Usually,  when  an  order-of-
magnitude calculation is made, the results are reliable to within about a factor of 10. If
a quantity increases in value by three orders of magnitude, this means that its value in-
creases by a factor of about 10

1 000. We use the symbol

# for “is on the order of.”

Thus,

0.008 6

# 10

0.002 1

# 10

720

# 10

The spirit of order-of-magnitude calculations, sometimes referred to as “guessti-

mates” or “ball-park figures,” is given in the following quotation: “Make an estimate
before every calculation, try a simple physical argument . . . before every derivation,
guess the answer to every puzzle.”

Inaccuracies caused by guessing too low for one

number are often canceled out by other guesses that are too high. You will find that
with practice your guesstimates become better and better. Estimation problems can
be fun to work as you freely drop digits, venture reasonable approximations for

S E C T I O N 1 . 6 • Estimates and Order-of-Magnitude Calculations

Example 1.4 Is He Speeding?

On an interstate highway in a rural region of Wyoming, a
car is traveling at a speed of 38.0 m/s. Is this car exceeding
the speed limit of 75.0 mi/h?

Solution We first convert meters to miles:

Now we convert seconds to hours:

Thus, the car is exceeding the speed limit and should slow
down.

What If?

What if the driver is from outside the U.S. and is

familiar with speeds measured in km/h? What is the speed
of the car in km/h?

Answer We can convert our final answer to the appropriate
units:

(85.0 mi/h)

1.609 km

1 mi

137 km/h

(2.36 ! 10

mi/s)

60 s

1 min

60 min

1 h

85.0 mi/h

(38.0 m/s)

1 mi

1 609 m

2.36 ! 10

mi/s

Figure 1.3 shows the speedometer of an automobile, with
speeds in both mi/h and km/h. Can you check the conver-
sion we just performed using this photograph?

Figure 1.3 The speedometer of a vehicle that
shows speeds in both miles per hour and kilome-
ters per hour.

Phil Boorman/Getty Images

E. Taylor and J. A. Wheeler, Spacetime Physics: Introduction to Special Relativity, 2nd ed., San Francisco,

W. H. Freeman & Company, Publishers, 1992, p. 20.

C H A P T E R 1 • Physics and Measurement

Example 1.5 Breaths in a Lifetime

Estimate the number of breaths taken during an average life
span.

Solution We  start  by  guessing  that  the  typical  life  span  is
about 70 years. The only other estimate we must make in this
example  is  the  average  number  of  breaths  that  a  person
takes  in  1 min.  This  number  varies,  depending  on  whether
the  person  is  exercising,  sleeping,  angry,  serene,  and  so
forth. To the nearest order of magnitude, we shall choose 10
breaths  per  minute  as  our  estimate  of  the  average.  (This  is
certainly closer to the true value than 1 breath per minute or
100 breaths per minute.) The number of minutes in a year is
approximately

Notice how much simpler it is in the expression above to
multiply 400 ! 25 than it is to work with the more accurate
365 ! 24. These approximate values for the number of days

6 ! 10

min

1 yr

400 days

1 yr

25 h

1 day

60 min

1 h

in a year and the number of hours in a day are close
enough for our purposes. Thus, in 70 years there will be
(70 yr)(6 ! 10

min/yr) $ 4 ! 10

min. At a rate of 10

breaths/min, an individual would take

in a lifetime, or on the order of 10

breaths.

What If?

What if the average life span were estimated as

80 years instead of 70? Would this change our final estimate?

Answer We could claim that (80 yr)(6 ! 10

min/yr) $

5 ! 10

min, so that our final estimate should be 5 ! 10

breaths. This is still on the order of 10

breaths, so an order-

of-magnitude  estimate  would  be  unchanged.  Furthermore,
80  years  is  14%  larger  than  70  years,  but  we  have  overesti-
mated the total time interval by using 400 days in a year in-
stead of 365 and 25 hours in a day instead of 24. These two
numbers  together  result  in  an  overestimate  of  14%,  which
cancels the effect of the increased life span!

4 ! 10

breaths

Example 1.6 It’s a Long Way to San Jose

Estimate the number of steps a person would take walking
from New York to Los Angeles.

Solution Without  looking  up  the  distance  between  these
two cities, you might remember from a geography class that
they are about 3 000 mi apart. The next approximation we
must make is the length of one step. Of course, this length
depends on the person doing the walking, but we can esti-
mate  that  each  step  covers  about  2  ft.  With  our  estimated
step size, we can determine the number of steps in 1 mi. Be-
cause  this  is  a  rough  calculation,  we  round  5  280 ft/mi  to
5 000  ft/mi.  (What  percentage  error  does  this  introduce?)
This conversion factor gives us

000 ft/mi

2 ft/step

500 steps/mi

Now we switch to scientific notation so that we can do the
calculation mentally:

So if we intend to walk across the United States, it will take
us on the order of ten million steps. This estimate is almost
certainly too small because we have not accounted for curv-
ing roads and going up and down hills and mountains.
Nonetheless, it is probably within an order of magnitude of
the correct answer.

7.5 ! 10

steps

# 10

steps

(3 ! 10

mi)(2.5 ! 10

steps/mi)

Example 1.7 How Much Gas Do We Use?

Estimate the number of gallons of gasoline used each year
by all the cars in the United States.
Solution Because there are about 280 million people in
the United States, an estimate of the number of cars in the
country is 100 million (guessing that there are between two
and three people per car). We also estimate that the average

distance each car travels per year is 10 000 mi. If we assume
a gasoline consumption of 20 mi/gal or 0.05 gal/mi, then
each car uses about 500 gal/yr. Multiplying this by the total
number of cars in the United States gives an estimated total

consumption of 5 ! 10

gal

# 10

gal.

unknown numbers, make simplifying assumptions, and turn the question around
into something you can answer in your head or with minimal mathematical manipu-
lation on paper. Because of the simplicity of these types of calculations, they can be
performed on a small piece of paper, so these estimates are often called “back-of-the-
envelope calculations.”

1.7 Significant Figures

When certain quantities are measured, the measured values are known only to within
the limits of the experimental uncertainty. The value of this uncertainty can depend
on various factors, such as the quality of the apparatus, the skill of the experimenter,
and the number of measurements performed. The number of

significant figures in a

measurement can be used to express something about the uncertainty.

As an example of significant figures, suppose that we are asked in a laboratory ex-

periment to measure the area of a computer disk label using a meter stick as a measur-
ing instrument. Let us assume that the accuracy to which we can measure the length of
the label is * 0.1 cm. If the length is measured to be 5.5 cm, we can claim only that its
length lies somewhere between 5.4 cm and 5.6 cm. In this case, we say that the mea-
sured value has two significant figures. Note that the significant figures include the first
estimated digit. Likewise, if the label’s width is measured to be 6.4 cm, the actual
value lies between 6.3 cm and 6.5 cm. Thus we could write the measured values as
(5.5 * 0.1) cm and (6.4 * 0.1) cm.

Now suppose we want to find the area of the label by multiplying the two measured

values. If we were to claim the area is (5.5 cm)(6.4 cm) $ 35.2 cm

, our answer would

be unjustifiable because it contains three significant figures, which is greater than the
number of significant figures in either of the measured quantities. A good rule of
thumb to use in determining the number of significant figures that can be claimed in a
multiplication or a division is as follows:

S E C T I O N 1 . 7 • Significant Figures

When multiplying several quantities, the number of significant figures in the final
answer is the same as the number of significant figures in the quantity having the
lowest number of significant figures. The same rule applies to division.

Applying this rule to the previous multiplication example, we see that the answer

for the area can have only two significant figures because our measured quantities
have only two significant figures. Thus, all we can claim is that the area is 35 cm

realizing that the value can range between (5.4 cm)(6.3 cm) $ 34 cm

and

(5.6 cm)(6.5 cm) $ 36 cm

Zeros may or may not be significant figures. Those used to position the decimal

point in such numbers as 0.03 and 0.007 5 are not significant. Thus, there are one
and two significant figures, respectively, in these two values. When the zeros come af-
ter other digits, however, there is the possibility of misinterpretation. For example,
suppose the mass of an object is given as 1 500 g. This value is ambiguous because we
do not know whether the last two zeros are being used to locate the decimal point or
whether they represent significant figures in the measurement. To remove this ambi-
guity, it is common to use scientific notation to indicate the number of significant fig-
ures. In this case, we would express the mass as 1.5 ! 10

g if there are two signifi-

cant figures in the measured value, 1.50 ! 10

g if there are three significant figures,

and 1.500 ! 10

g if there are four. The same rule holds for numbers less than 1, so

that 2.3 ! 10

has two significant figures (and so could be written 0.000 23) and

2.30 ! 10

has three significant figures (also written 0.000 230). In general,

a sig-

nificant figure in a measurement is a reliably known digit (other than a zero
used to locate the decimal point) or the first estimated digit.

For addition and subtraction, you must consider the number of decimal places

when you are determining how many significant figures to report:

When numbers are added or subtracted, the number of decimal places in the result
should equal the smallest number of decimal places of any term in the sum.

▲

PITFALL PREVENTION

1.6 Read Carefully

Notice  that  the  rule  for  addition
and subtraction is different from
that  for  multiplication  and  divi-
sion.  For  addition  and  subtrac-
tion, the important consideration
is  the  number  of  decimal  places,
not  the  number  of  significant
figures.

For example, if we wish to compute 123 ' 5.35, the answer is 128 and not 128.35. If we
compute the sum 1.000 1 ' 0.000 3 $ 1.000 4, the result has five significant figures,
even though one of the terms in the sum, 0.000 3, has only one significant figure. Like-
wise, if we perform the subtraction 1.002 " 0.998 $ 0.004, the result has only one sig-
nificant figure even though one term has four significant figures and the other has
three. In this book,

most of the numerical examples and end-of-chapter problems

will yield answers having three significant figures. When carrying out estimates we
shall typically work with a single significant figure.

If the number of significant figures in the result of an addition or subtraction

must be reduced, there is a general rule for rounding off numbers, which states that
the last digit retained is to be increased by 1 if the last digit dropped is greater than
5. If the last digit dropped is less than 5, the last digit retained remains as it is. If the
last digit dropped is equal to 5, the remaining digit should be rounded to the near-
est even number. (This helps avoid accumulation of errors in long arithmetic
processes.)

A technique for avoiding error accumulation is to delay rounding of numbers in a

long calculation until you have the final result. Wait until you are ready to copy the fi-
nal answer from your calculator before rounding to the correct number of significant
figures.

C H A P T E R 1 • Physics and Measurement

Quick Quiz 1.4

Suppose you measure the position of a chair with a meter

stick and record that the center of the seat is 1.043 860 564 2 m from a wall. What
would a reader conclude from this recorded measurement?

Example 1.8 Installing a Carpet

A carpet is to be installed in a room whose length is mea-
sured to be 12.71 m and whose width is measured to be
3.46 m. Find the area of the room.

Solution If you multiply 12.71 m by 3.46 m on your calcula-
tor, you will see an answer of 43.976 6 m

. How many of these

numbers should you claim? Our rule of thumb for multiplica-
tion tells us that you can claim only the number of significant
figures in your answer as are present in the measured quan-
tity having the lowest number of significant figures. In this ex-
ample, the lowest number of significant figures is three in

3.46 m, so we should express our final answer as

44.0 m

The three fundamental physical quantities of mechanics are length, mass, and time,
which in the SI system have the units meters (m), kilograms (kg), and seconds (s), re-
spectively. Prefixes indicating various powers of ten are used with these three basic
units.

The

density of a substance is defined as its mass per unit volume. Different sub-

stances have different densities mainly because of differences in their atomic masses
and atomic arrangements.

The method of

dimensional analysis is very powerful in solving physics problems.

Dimensions can be treated as algebraic quantities. By making estimates and perform-
ing order-of-magnitude calculations, you should be able to approximate the answer to
a problem when there is not enough information available to completely specify an ex-
act solution.

When you compute a result from several measured numbers, each of which has a

certain accuracy, you should give the result with the correct number of

significant fig-

ures. When multiplying several quantities, the number of significant figures in the

S U M M A R Y

Take a practice test for

this chapter by clicking on
the Practice Test link at
http://www.pse6.com.

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