100 yd $ 300 ft), and determine the diameter of the

nucleus in millimeters. (b) The atom is how many times

larger in volume than its nucleus?

36. The nearest stars to the Sun are in the Alpha Centauri

multiple-star system, about 4.0 ! 1013 km away. If the Sun,

with a diameter of 1.4 ! 109 m, and Alpha Centauri A are

both represented by cherry pits 7.0 mm in diameter, how

far apart should the pits be placed to represent the Sun

and its neighbor to scale?

The diameter of our disk-shaped galaxy, the Milky Way, is

about 1.0 ! 105 lightyears (ly). The distance to Messier 31,

which is Andromeda, the spiral galaxy nearest to the Milky

Way, is about 2.0 million ly. If a scale model represents the

Milky Way and Andromeda galaxies as dinner plates 25 cm

in diameter, determine the distance between the two plates.

38. The mean radius of the Earth is 6.37 ! 106 m, and that of

the Moon is 1.74 ! 108 cm. From these data calculate

(a) the ratio of the Earth’s surface area to that of the

Moon and (b) the ratio of the Earth’s volume to that of

the Moon. Recall that the surface area of a sphere is 4#r 2

and the volume of a sphere is

One cubic meter (1.00 m3) of aluminum has a mass

of 2.70 ! 103 kg, and 1.00 m3 of iron has a mass of

7.86 ! 103 kg. Find the radius of a solid aluminum sphere

that will balance a solid iron sphere of radius 2.00 cm on

an equal-arm balance.

40. Let &Al represent the density of aluminum and & Fe that of

iron. Find the radius of a solid aluminum sphere that balances

a solid iron sphere of radius r Fe on an equal-arm

balance.

Section 1.6 Estimates and Order-of-Magnitude

Calculations

Estimate the number of Ping-Pong balls that would fit

into a typical-size room (without being crushed). In your

solution state the quantities you measure or estimate and

the values you take for them.

42. An automobile tire is rated to last for 50 000 miles. To an

order of magnitude, through how many revolutions will it

turn? In your solution state the quantities you measure or

estimate and the values you take for them.

43. Grass grows densely everywhere on a quarter-acre plot of

land. What is the order of magnitude of the number of

blades of grass on this plot? Explain your reasoning. Note

that 1 acre $ 43 560 ft2.

44. Approximately how many raindrops fall on a one-acre lot

during a one-inch rainfall? Explain your reasoning.

45. Compute the order of magnitude of the mass of a bathtub

half full of water. Compute the order of magnitude of the

mass of a bathtub half full of pennies. In your solution list

the quantities you take as data and the value you measure

or estimate for each.

46. Soft drinks are commonly sold in aluminum containers. To

an order of magnitude, how many such containers are

thrown away or recycled each year by U.S. consumers?

41.

39.

43

#r 3.

37.

How many tons of aluminum does this represent? In your

solution state the quantities you measure or estimate and

the values you take for them.

To an order of magnitude, how many piano tuners are in

New York City? The physicist Enrico Fermi was famous for

asking questions like this on oral Ph.D. qualifying examinations.

His own facility in making order-of-magnitude calculations

is exemplified in Problem 45.48.

Section 1.7 Significant Figures

48. A rectangular plate has a length of (21.3 * 0.2) cm and a

width of (9.8 * 0.1) cm. Calculate the area of the plate, including

its uncertainty.

49. The radius of a circle is measured to be (10.5 * 0.2)m.

Calculate the (a) area and (b) circumference of the circle

and give the uncertainty in each value.

50. How many significant figures are in the following numbers?

(a) 78.9 * 0.2 (b) 3.788 ! 109 (c) 2.46 ! 10"6

(d) 0.005 3.

51. The radius of a solid sphere is measured to be

(6.50 * 0.20) cm, and its mass is measured to be

(1.85 * 0.02) kg. Determine the density of the sphere in

kilograms per cubic meter and the uncertainty in the

density.

52. Carry out the following arithmetic operations: (a) the sum

of the measured values 756, 37.2, 0.83, and 2.5; (b) the

product 0.003 2 ! 356.3; (c) the product 5.620 ! #.

53. The tropical year, the time from vernal equinox to the next

vernal equinox, is the basis for our calendar. It contains

365.242 199 days. Find the number of seconds in a tropical

year.

54. A farmer measures the distance around a rectangular field.

The length of the long sides of the rectangle is found to

be 38.44 m, and the length of the short sides is found to

be 19.5 m. What is the total distance around the field?

55. A sidewalk is to be constructed around a swimming pool

that measures (10.0 * 0.1)m by (17.0 * 0.1) m. If the sidewalk

is to measure (1.00 * 0.01)m wide by (9.0 * 0.1) cm

thick, what volume of concrete is needed, and what is the

approximate uncertainty of this volume?

Additional Problems

56. In a situation where data are known to three significant

digits, we write 6.379 m $ 6.38 m and 6.374 m $ 6.37m.

When a number ends in 5, we arbitrarily choose to write

6.375 m $ 6.38 m. We could equally well write 6.375 m $

6.37 m, “rounding down” instead of “rounding up,” because

we would change the number 6.375 by equal increments

in both cases. Now consider an order-of-magnitude

estimate, in which we consider factors rather than increments.

We write 500 m # 103 m because 500 differs from

100 by a factor of 5 while it differs from 1 000 by only a factor

of 2. We write 437 m # 103 m and 305 m #102 m.

What distance differs from 100 m and from 1 000 m

by equal factors, so that we could equally well choose to

represent its order of magnitude either as #102 m or as

#103 m?

57. For many electronic applications, such as in computer

chips, it is desirable to make components as small as possible

to keep the temperature of the components low and to

increase the speed of the device. Thin metallic coatings

(films) can be used instead of wires to make electrical connections.

Gold is especially useful because it does not oxidize

readily. Its atomic mass is 197 u. A gold film can be

no thinner than the size of a gold atom. Calculate the

minimum coating thickness, assuming that a gold atom occupies

a cubical volume in the film that is equal to the volume

it occupies in a large piece of metal. This geometric

model yields a result of the correct order of magnitude.

58. The basic function of the carburetor of an automobile is to

“atomize” the gasoline and mix it with air to promote

rapid combustion. As an example, assume that 30.0 cm3 of

gasoline is atomized into N spherical droplets, each with a

radius of 2.00!10"5 m. What is the total surface area of

these N spherical droplets?

The consumption of natural gas by a company satisfies

the empirical equation V $ 1.50t ' 0.008 00t 2, where

V is the volume in millions of cubic feet and t the time in

months. Express this equation in units of cubic feet and

seconds. Assign proper units to the coefficients. Assume a

month is equal to 30.0 days.

60. In physics it is important to use mathematical approximations.

Demonstrate that for small angles (+20°)

tan , % sin , % , $ #,-/180°

where , is in radians and ,- is in degrees. Use a calculator

to find the largest angle for which tan , may be approximated

by sin , if the error is to be less than 10.0%.

A high fountain of water is located at the center of a circular

pool as in Figure P1.61. Not wishing to get his feet wet,

61.

59.

a student walks around the pool and measures its circumference

to be 15.0 m. Next, the student stands at the edge

of the pool and uses a protractor to gauge the angle of elevation

of the top of the fountain to be 55.0°. How high is

the fountain?

62. Collectible coins are sometimes plated with gold to enhance

their beauty and value. Consider a commemorative

quarter-dollar advertised for sale at $4.98. It has a diameter

of 24.1 mm, a thickness of 1.78 mm, and is completely

covered with a layer of pure gold 0.180%m thick. The volume

of the plating is equal to the thickness of the layer

times the area to which it is applied. The patterns on the

faces of the coin and the grooves on its edge have a negligible

effect on its area. Assume that the price of gold is

$10.0 per gram. Find the cost of the gold added to the

coin. Does the cost of the gold significantly enhance the

value of the coin?

There are nearly # ! 107 s in one year. Find the percentage

error in this approximation, where “percentage error’’

is defined as

64. Assume that an object covers an area A and has a uniform

height h. If its cross-sectional area is uniform over its

height, then its volume is given by V $ Ah. (a) Show that

V $ Ah is dimensionally correct. (b) Show that the volumes

of a cylinder and of a rectangular box can be written

in the form V $ Ah, identifying A in each case. (Note that

A, sometimes called the “footprint” of the object, can have

any shape and the height can be replaced by average

thickness in general.)

65. A child loves to watch as you fill a transparent plastic bottle

with shampoo. Every horizontal cross-section is a circle,

but the diameters of the circles have different values,

so that the bottle is much wider in some places than others.

You pour in bright green shampoo with constant volume

flow rate 16.5 cm3/s. At what rate is its level in the

bottle rising (a) at a point where the diameter of the bottle

is 6.30 cm and (b) at a point where the diameter is

1.35 cm?

66. One cubic centimeter of water has a mass of 1.00 ! 10"3 kg.

(a) Determine the mass of 1.00 m3 of water. (b) Biological

substances are 98% water. Assume that they have the same

density as water to estimate the masses of a cell that has a diameter

of 1.0%m, a human kidney, and a fly. Model the kidney

as a sphere with a radius of 4.0 cm and the fly as a cylinder

4.0 mm long and 2.0 mm in diameter.

Assume there are 100 million passenger cars in the United

States and that the average fuel consumption is 20 mi/gal of

gasoline. If the average distance traveled by each car is

10 000 mi/yr, how much gasoline would be saved per year if

average fuel consumption could be increased to 25 mi/gal?

68. A creature moves at a speed of 5.00 furlongs per fortnight

(not a very common unit of speed). Given that

1 furlong $ 220 yards and 1 fortnight $ 14 days, determine

the speed of the creature in m/s. What kind of creature

do you think it might be?

69. The distance from the Sun to the nearest star is about

4 ! 1016 m. The Milky Way galaxy is roughly a disk of diameter

#1021 m and thickness#1019 m. Find the order

of magnitude of the number of stars in the Milky Way.

Assume the distance between the Sun and our nearest

neighbor is typical.

70. The data in the following table represent measurements

of the masses and dimensions of solid cylinders of aluminum,

copper, brass, tin, and iron. Use these data to

calculate the densities of these substances. Compare your

results for aluminum, copper, and iron with those given

in Table 1.5.

Mass Diameter Length

Substance (g) (cm) (cm)

Aluminum 51.5 2.52 3.75

Copper 56.3 1.23 5.06

Brass 94.4 1.54 5.69

Tin 69.1 1.75 3.74

Iron 216.1 1.89 9.77

71. (a) How many seconds are in a year? (b) If one micrometeorite

(a sphere with a diameter of 1.00 ! 10"6 m)

strikes each square meter of the Moon each second, how

many years will it take to cover the Moon to a depth of

1.00 m? To solve this problem, you can consider a cubic

box on the Moon 1.00 m on each edge, and find how long

it will take to fill the box.