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
Section 1.6 Estimates and Order-of-Magnitude
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?
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
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
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
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?
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
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,
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
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
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.