Chapter 11 Gravity

Chapter 11
Gravity
Conceptual Problems
1 • True or false:
(a) For Kepler’s law of equal areas to be valid, the force of gravity must vary
inversely with the square of the distance between a given planet and the
Sun.
(b) The planet closest to the Sun has the shortest orbital period.
(c) Venus’s orbital speed is larger than the orbital speed of Earth.
(d) The orbital period of a planet allows accurate determination of that planet’s
mass.
(a) False. Kepler’s law of equal areas is a consequence of the fact that the
gravitational force acts along the line joining two bodies but is independent
of the manner in which the force varies with distance.
(b) True. The periods of the planets vary with the three-halves power of their
distances from the Sun. So the shorter the distance from the Sun, the shorter the
period of the planet’s motion.
(c) True. Setting up a proportion involving the orbital speeds of the two planets in
terms of their orbital periods and mean distances from the Sun (see Table 11-1)
shows that v = 1. 17v
.
Venus
Earth
(d) False. The orbital period of a planet is independent of the planet’s mass.
3 • During what season in the northern hemisphere does Earth attain its
maximum orbital speed about the Sun? What season is related to its minimum
orbital speed? Hint: Earth is at perihelion in early January.
Determine the Concept Earth is closest to the Sun during winter in the northern
hemisphere. This is the time of fastest orbital speed. Summer would be the time
for minimum orbital speed.
7 • At the surface of the moon, the acceleration due to the gravity of the
moon is a. At a distance from the center of the moon equal to four times the
radius of the moon, the acceleration due to the gravity of the moon is (a) 16a, (b)
a/4, (c) a/3, (d) a/16, (e) None of the above.
Picture the Problem The acceleration due to gravity varies inversely with the
square of the distance from the center of the moon.
221

222
Chapter 11
Express the dependence of the
acceleration due to the gravity of the
moon on the distance from its center:
Express the dependence of the
acceleration due to the gravity of the
moon at its surface on its radius:
Divide the first of these expressions
by the second to obtain:
Solving for a′ and simplifying yields:
1
a' ∝ 2
r
1
a ∝
R
a'
=
a
2
M
R
r
R
=
r
2
M
2
R
a' a a 16
( R )
1
2
2
M
M = =
2
2
4 M
and (d) is correct.
11 •• Suppose the escape speed from a planet was only slightly larger than
the escape speed from Earth, yet the planet is considerably larger than Earth. How
would the planet’s (average) density compare to Earth’s (average) density? (a) It
must be denser. (b) It must be less dense. (c) It must be the same density. (d) You
cannot determine the answer based on the data given.
Picture the Problem The densities of the planets are related to the escape speeds
from their surfaces through v = 2GM
R .
The escape speed from the planet is
given by:
The escape speed from Earth is given
by:
Expressing the ratio of the escape
speed from the planet to the escape
speed from Earth and simplifying
yields:
Because vplanet ≈ vEarth:
e
v =
planet
v =
v
v
Earth
planet
Earth
1 ≈
=
R
R
Earth
planet
2GM
R
2GM
R
planet
Earth
2GM
R
planet
Earth
planet
2GM
R
M
M
Earth
planet
Earth
Earth
planet
=
a
R
R
Earth
planet
M
M
planet
Earth

Squaring both sides of the equation
yields:
R
1 ≈
R
Earth
planet
M
M
planet
Earth
Gravity
Express Mplanet and MEarth in terms of their densities and simplify to obtain:
R
1 ≈
R
Earth
planet
ρ
ρ
planet
Earth
V
V
planet
Earth
R
=
R
Earth
planet
planet
Earth
Solving for the ratio of the densities
yields:
Because the planet is considerably
larger than Earth:
ρ
ρ
V
V
planet
Earth
R
=
R
ρ
ρ
ρ
ρ
planet
Earth
Earth
planet
ρ
ρ
R
≈
R
planet

224
Chapter 11
Using Kepler’s third law, relate the
period of the Sun T to its mean
distance R0 from the center of the
galaxy:
Substitute numerical values and evaluate MG/Ms:
M
M
G
s
galaxy
T
4π
M 4π
R
2
2 3
2
=
GM G
3 G R0
⇒
M S
0 =
2
GM ST
15
2⎛
4 9.
461×
10 m ⎞
4π
⎜
⎜3.00×
10 c ⋅ y×
c y ⎟
⎝
⋅
=
⎠
2
⎛ −11
N ⋅ m ⎞
30 ⎛
6
⎜
⎜6.
6742×
10 ( 1.
99×
10 kg)
⎜250×
10 y×
3.
156×
10
2
kg ⎟
⎝
⎠
⎝
11
≈ 1.
1×
10 ⇒ M ≈
11
1.
1×
10 M
Kepler’s Laws
25 •• [SSM] One of the so-called ″Kirkwood gaps″ in the asteroid belt
occurs at an orbital radius at which the period of the orbit is half that of Jupiter’s.
The reason there is a gap for orbits of this radius is because of the periodic pulling
(by Jupiter) that an asteroid experiences at the same place in its orbit every other
orbit around the sun. Repeated tugs from Jupiter of this kind would eventually
change the orbit of such an asteroid. Therefore, all asteroids that would otherwise
have orbited at this radius have presumably been cleared away from the area due
to this resonance phenomenon. How far from the Sun is this particular 2:1
resonance ″Kirkwood″ gap?
Picture the Problem The period of an orbit is related to its semi-major axis (for
circular orbits this distance is the orbital radius). Because we know the orbital
periods of Jupiter and a hypothetical asteroid in the Kirkwood gap, we can use
Kepler’s third law to set up a proportion relating the orbital periods and average
distances of Jupiter and the asteroid from the Sun from which we can obtain an
expression for the orbital radius of an asteroid in the Kirkwood gap.
Use Kepler’s third law to relate
Jupiter’s orbital period to its mean
distance from the Sun:
T
s
2
Jupiter
2
4π
=
GM
Sun
r
3
3
Jupiter
7
s
y
⎞
⎟
⎠
2

Use Kepler’s third law to relate the
orbital period of an asteroid in the
Kirkwood gap to its mean distance
from the Sun:
Dividing the second of these
equations by the first and simplifying
yields:
Solving for rKirkwood yields:
Because the period of the orbit of an
asteroid in the Kirkwood gap is half
that of Jupiter’s:
T
2
Kirkwood
T
2
Kirkwood
2
TJupiter
2
4π
=
GM
Sun
2
4π
GM
=
4π
GM
Sun
2
r
Sun
3
Kirkwood
r
3
Kirkwood
r
3
Jupiter
Gravity
r
=
2 3
Kirkwood
Kirkwood Jupiter ⎟
Jupiter
⎟
⎛ ⎞
⎜
T
= r
⎜ T
r
Kirkwood
=
=
⎝
10 ( 77.
8×
10 m)
4.
90×
10
11
⎠
⎛ 1
2 T
⎜
⎝ T
m
3
Kirkwood
3
rJupiter
Jupiter
Jupiter
⎞
⎟
⎠
11 1AU
= 4.
90×
10 m×
1.50×
10
= 3.
27 AU
Remarks: There are also significant Kirkwood gaps at 3:1, 5:2, and 7:3 and
resonances at 2.5 AU, 2.82 AU, and 2.95 AU.
29 •• Kepler determined distances in the Solar System from his data. For
example, he found the relative distance from the Sun to Venus (as compared to
the distance from the Sun to Earth) as follows. Because Venus’s orbit is closer to
the Sun than is Earth’s orbit, Venus is a morning or evening star—its position in
the sky is never very far from the Sun (Figure 11-24). If we suppose the orbit of
Venus is a perfect circle, then consider the relative orientation of Venus, Earth,
and the Sun at maximum extension, that is when Venus is farthest from the Sun in
the sky. (a) Under this condition, show that angle b in Figure 11-24 is 90º. (b) If
the maximum elongation angle a between Venus and the Sun is 47º, what is the
distance between Venus and the Sun in AU? (c) Use this result to estimate the
length of a Venusian ″year.″
Picture the Problem We can use a property of lines tangent to a circle and radii
drawn to the point of contact to show that b = 90°. Once we’ve established that b
is a right angle we can use the definition of the sine function to relate the distance
from the Sun to Venus to the distance from the Sun to Earth.
r
2 3
11
m
225

226
Chapter 11
(a) The line from Earth to Venus' orbit is tangent to the orbit of Venus at the point
of maximum extension. Venus will appear closer to the sun in Earth’s sky when it
passes the line drawn from Earth and tangent to its orbit. Hence b = 90°
(b) Using trigonometry, relate the
distance from the sun to Venus d
to the angle a:
Substitute numerical values and
evaluate dSV
:
(c) Use Kepler’s third law to relate
Venus’s orbital period to its mean
distance from the Sun:
Use Kepler’s third law to relate
Earth’s orbital period to its mean
distance from the Sun:
Dividing the first of these equations
by the second and simplifying yields:
Solving for TVenus
yields:
SV
Using the result from Part (b) yields:
dSV
a = ⇒ dSV
= d sin a
d
sin SE
d
T
T
SV
=
=
2
Venus
2
Earth
T
T
T
2
Venus
2
Earth
SE
( 1.
00AU)
0.
73AU
2
4π
=
GM
2
4π
=
GM
Sun
Sun
2
4π
GM
=
4π
GM
sin 47°
=
r
r
Sun
2
Sun
3
Venus
3
Earth
r
3
Venus
r
3
Earth
3 2
Venus
Venus Earth ⎟
Earth
⎟
⎛ r ⎞
= T ⎜
r
Venus
=
⎝
( 1.
00 y)
0.
63 y
⎠
r
=
r
0.
731AU
3
Venus
3
Earth
⎛ 0.
731AU
⎞
⎜ ⎟
⎝ 1.
00 AU ⎠
Remarks: The correct distance from the sun to Venus is closer to 0.723 AU.
Newton’s Law of Gravity
31 • Jupiter’s satellite Europa orbits Jupiter with a period of 3.55 d at an
average orbital radius of 6.71 × 10 8 m. (a) Assuming that the orbit is circular,
determine the mass of Jupiter from the data given. (b) Another satellite of Jupiter,
Callisto, orbits at an average radius of 18.8 × 10 8 m with an orbital period of
T
=
3 2

Gravity
16.7 d. Show that these data are consistent with an inverse square force law for
gravity (Note: DO NOT use the value of G anywhere in Part (b)).
Picture the Problem While we could apply Newton’s law of gravitation and
second law of motion to solve this problem from first principles, we’ll use
Kepler’s third law (derived from these laws) to find the mass of Jupiter in Part (a).
In Part (b) we can compare the ratio of the centripetal accelerations of Europa and
Callisto to show that these data are consistent with an inverse square law for
gravity.
(a) Assuming a circular orbit, apply
Kepler’s third law to the motion of
Europa to obtain:
Substitute numerical values and evaluate MJ:
2
8
π ( 6.71×
10 m)
−11
2 2 ⎛
( 6.
673×
10 N ⋅ m /kg ) ⎜3.55d
T
2
E
2
2
4π
3 4π
= RE
⇒ M J = R
GM
GT
3
4 27
M J =
= 1.
90 × 10
2
24 h 3600s
⎞
× × ⎟
⎝ d h ⎠
Note that this result is in excellent agreement with the accepted value of
1.902×10 27 kg.
(b) Express the centripetal
acceleration of both of the moons to
obtain:
Using this result, express the
centripetal accelerations of Europa
and Callisto:
Divide the first of these equations by
the second and simplify to obtain:
Substitute for the periods of Callisto
and Europa using Kepler’s third
law to obtain:
J
2
2
E
3
E
kg
227
⎛ 2πR
⎞
2 ⎜ ⎟ 2
v T 4π
R
acentripetal
= =
⎝ ⎠
= 2
R R T
where R and T are the radii and periods
of their motion.
a
4π
R
4π
R
2
2
E = 2
TE
E and aC
= C
2
TC
a
a
a
a
E
C
E
C
4π
R
T
=
4π
T
CR
=
CR
2
E
2
E
2
RC
2
C
3
C
3
E
R
R
E
C
T
=
T
2
C
2
E
R
=
R
2
C
2
E
R
R
E
C

228
Chapter 11
This result, together with the fact that the gravitational force is directly
proportional to the acceleration of the moons, demonstrates that the gravitational
force varies inversely with the square of the distance.
33 • The mass of Saturn is 5.69 × 10 26 kg. (a) Find the period of its moon
Mimas, whose mean orbital radius is 1.86 × 10 8 m. (b) Find the mean orbital
radius of its moon Titan, whose period is 1.38 × 10 6 s.
Picture the Problem While we could apply Newton’s law of gravitation and
second law of motion to solve this problem from first principles, we’ll use
Kepler’s third law (derived from these laws) to find the period of Mimas and to
relate the periods of the moons of Saturn to their mean distances from its center.
(a) Using Kepler’s third law, relate
the period of Mimas to its mean
distance from the center of Saturn:
Substitute numerical values and evaluate TM:
T
T
2
M
2
8 3
π ( 1.
86×
10 m)
11 2 2
26
( 6.
6726×
10 N ⋅ m /kg )( 5.
69×
10 kg)
2
4π
3
= rM
⇒ T
GM
4 4
M = = 8.
18×
10
−
(b) Using Kepler’s third law, relate
the period of Titan to its mean
distance from the center of Saturn:
Substitute numerical values and evaluate rT:
r
T
=
3
T
4π
S
M
2
2
3
T = rT
⇒
GM S
3 r T =
=
s ≈
2
4π
GM
2
TT
GM
2
4π
S
r
3
M
22.7 h
6 2
−11
2 2
26
( 1. 38×
10 s)
( 6.
6726×
10 N ⋅ m /kg )( 5.
69 × 10 kg)
9
= 1.
22 × 10 m
2
4π
41 •• A superconducting gravity meter can measure changes in gravity of
the order Δg/g = 1.00 × 10 –11 . (a) You are hiding behind a tree holding the meter,
and your 80-kg friend approaches the tree from the other side. How close to you
can your friend get before the meter detects a change in g due to his presence? (b)
You are in a hot air balloon and are using the meter to determine the rate of ascent
(assume the balloon has constant acceleration). What is the smallest change in
altitude that results in a detectable change in the gravitational field of Earth?
S

Gravity
Picture the Problem We can determine the maximum range at which an object
with a given mass can be detected by substituting the equation for the
gravitational field in the expression for the resolution of the meter and solving for
the distance. Differentiating g(r) with respect to r, separating variables to obtain
dg/g, and approximating Δr with dr will allow us to determine the vertical change
in the position of the gravity meter in Earth’s gravitational field is detectable.
(a) Earth’s gravitational field is
given by:
Express the gravitational field due to
the mass m (assumed to be a point
mass) of your friend and relate it to
the resolution of the meter:
Solving for r yields:
GM
g E =
R
g
() r
r = R
E
2
E
Gm
= = 1.
00×
10
2
r
−11
GM
= 1.
00×
10
R
E
1.
00×
10
M
Substitute numerical values and
11
6 1.
00×
10
evaluate r: ( ) ( 80kg)
r = 6.37× 10 m
24
(b) Differentiate g(r) and simplify to
obtain:
=
7.
37m
E
11
m
−11
E
2
E
g
E
5.98×
10
kg
dg − 2Gm
2 ⎛ Gm ⎞ 2
= = − = − g
3 ⎜ 2 ⎟
dr r r ⎝ r ⎠ r
Separate variables to obtain: dg dr −11
= −2
= 10
g r
Approximating dr with Δr, evaluate
Δr with r = RE:
Δr
= −
1
2
−11
6
( 1.
00×
10 )( 6.
37×
10 m)
= 31.
9μm
43 •• Earth’s radius is 6370 km and the moon’s radius is
1738 km. The acceleration of gravity at the surface of the moon is 1.62 m/s 2 .
What is the ratio of the average density of the moon to that of Earth?
Picture the Problem We can use the definitions of the gravitational fields at the
surfaces of Earth and the moon to express the accelerations due to gravity at these
locations in terms of the average densities of Earth and the moon. Expressing the
ratio of these accelerations will lead us to the ratio of the densities.
229

230
Chapter 11
Express the acceleration due to
gravity at the surface of Earth in
terms of Earth’s average density:
The acceleration due to gravity at the
surface of the moon in terms of the
moon’s average density is:
g
E
M
GM
=
R
=
4
3
E
2
E
Gρ
π R
4
3
E
Gρ
V
=
R
M
E
g = Gρ
π R
E E
2
E
M
Gρ
=
Divide the second of these equations g M ρ M RM
ρM
gM
RE
= ⇒ =
by the first to obtain: g E ρ E RE
ρE
gE
RM
Substitute numerical values and
ρ M
evaluate :
ρ
E
Gravitational Potential Energy
ρM
ρ
E
=
=
4
E 3
2
RE
π R
2
6
( 1.62m/s
)( 6.37×
10 m)
2
6
( 9.81m/s
)( 1.738×
10 m)
0.
605
47 • Knowing that the acceleration of gravity on the moon is 0.166 times
that on Earth and that the moon’s radius is 0.273RE, find the escape speed for a
projectile leaving the surface of the moon.
Picture the Problem The escape speed from the moon is given by
v e, m = 2GM m Rm
, where Mm and Rm represent the mass and radius of the moon,
respectively.
Express the escape speed from the
moon:
Because g m = 0. 166gE
and
R = 0. 273R
:
m
E
2GM
m
v e.m = =
Rm
v =
Substitute numerical values and evaluate ve,m:
v
e.m
=
2
e.m
2g
m
R
m
( 0.
166g
)( 0.
273 )
2 R
2
6
( 0.
166)(
9.
81m/s
)( 0.
273)(
6.
371×
10 m)
= 2.
38 km/s
51 •• An object is dropped from rest from a height of 4.0 × 10 6 m above the
surface of Earth. If there is no air resistance, what is its speed when it strikes
Earth?
E
E
3
E

Gravity
Picture the Problem Let the zero of gravitational potential energy be at infinity
and let m represent the mass of the object. We’ll use conservation of energy to
relate the initial potential energy of the object-Earth system to the final potential
and kinetic energies.
Use conservation of energy to relate
the initial potential energy of the
system to its energy as the object is
about to strike Earth:
Express the potential energy of the
object-Earth system when the object
is at a distance r from the surface of
Earth:
Substitute in equation (1) to obtain:
Solving for v yields:
Substitute numerical values and evaluate v:
v =
Gravitational Orbits
2
231
K f − Ki
+ U f −U
i = 0
or, because Ki = 0,
K ( RE
) + U ( RE
) −U
( RE
+ h)
= 0 (1)
where h is the initial height above
Earth’s surface.
U
() r
mv
GM
= −
r
GM
E
m
m
GM
m
1 2 E
E
2 − + =
RE
RE
+ h
v =
=
⎛ GM
2 ⎜
⎝ RE
E
GM ⎞ E − ⎟
RE
+ h ⎠
⎛ h ⎞
2gR
⎜
⎟
E
⎝ RE
+ h ⎠
2
6
6
( 9.81m/s
)( 6.37×
10 m)(
4.
0×
10 m)
= 6.
9km/s
6.37 × 10
6
m + 4.
0×
10
59 •• Many satellites orbit Earth with maximum altitudes of
1000 km or less. Geosynchronous satellites, however, orbit at an altitude of
35 790 km above Earth’s surface. How much more energy is required to launch a
500-kg satellite into a geosynchronous orbit than into an orbit 1000 km above the
surface of Earth?
Picture the Problem We can express the energy difference between these two
orbits in terms of the total energy of a satellite at each elevation. The application
of Newton’s second law to the force acting on a satellite will allow us to express
6
m
0

232
Chapter 11
the total energy of each satellite as a function of its mass, the radius of Earth, and
its orbital radius.
Express the energy difference: Δ E = Egeo
− E1000
(1)
Express the total energy of an
orbiting satellite:
Apply Newton’s second law to a
satellite to relate the gravitational
force to the orbital speed:
Solving for v 2 yields:
Substitute in equation (2) to obtain:
Etot
= K + U
2 GM Em
1 = 2 mv −
R
where R is the orbital radius.
F
v
E
radial
2 =
tot
=
GM
=
R
gR
R
Substituting in equation (1) and simplifying yields:
1
2
2
E
E
2
gR
m
R
2
m v
= m
R
2
2
2
mgR
⎛ ⎞
E mgRE
mgRE
⎜
1 1
ΔE
= − + =
− ⎟
2R
⎜ ⎟
geo 2R1000
2 ⎝ R1000
Rgeo
⎠
2
mgR ⎛ 1
1 ⎞
E = ⎜
−
⎟
2 ⎝ RE
+ 1000 km RE
+ 35 790 km ⎠
Substitute numerical values and evaluate ΔE:
ΔE =
2
E
(2)
2
gREm
mgR
− = −
R 2R
6 2
( 500 kg)(
9.81N
/ kg)(
6.37 × 10 m)
⎛ 1
1 ⎞
⎜
−
⎟ = 11.
1GJ
The Gravitational Field ( g r )
2
⎜
⎝ 7.
37 × 10
6
m
4.
22×
107 m ⎟
⎠
63 •• A point particle of mass m is on the x axis at x = L and an identical
point particle is on the y axis at y = L. (a) What is the direction of the
gravitational field at the origin? (b) What is the magnitude of this field?
2
E

Gravity
Picture the Problem We can use the definition of the gravitational field due to a
point mass to find the x and y components of the field at the origin and then add
these components to find the resultant field. We can find the magnitude of the
field from its components using the Pythagorean theorem.
(a) The gravitational field at the
origin is the sum of its x and y
components:
Express the gravitational field due to
the point mass at x = L:
Express the gravitational field due to
the point mass at y = L:
Substitute in equation (1) to obtain:
r r r
g = g + g
(1)
r
g
r
g
x =
y =
x
y
Gm
iˆ
2
L
Gm
2
L
(b) The magnitude of g r is given by: 2 2
= g x + g y
Substitute for gx and gy and simplify
ˆj
r r r Gm
g g g iˆ
Gm
=
ˆ
x + y = + j ⇒ the
2 2
L L
direction of the gravitational field is
along a line at 45° above the +x axis.
⎛ Gm ⎞ ⎛ Gm ⎞ Gm
to obtain: g = ⎜
= 2 2 ⎟ + ⎜ 2 ⎟
2
⎝ L ⎠ ⎝ L ⎠ L
r
67 ••• A nonuniform thin rod of length L lies on the x axis. One end of the
rod is at the origin, and the other end is at x = L. The rod’s mass per unit length λ
varies as λ = Cx, where C is a constant. (Thus, an element of the rod has mass
dm = λ dx.) (a) What is the total mass of the rod? (b) Find the gravitational field
due to the rod on the x axis at x = x0, where x0 > L.
Picture the Problem We can find the mass of the rod by integrating dm over its
length. The gravitational field at x0 > L can be found by integrating g at x
r
d 0 over
the length of the rod.
(a) The total mass of the stick is
given by:
Substitute for λ and evaluate the
integral to obtain:
g r
M =
L
∫
0
λ dx
L
0
2
M = C∫
xdx =
1
2
CL
2
2
233

234
Chapter 11
(b) Express the gravitational field
due to an element of the stick of
mass dm:
Integrate this expression over the
r
dg
= −
= −
Gdm
2 ( x − x)
( x − x)
0
GCxdx
( ) i 2
x − x
0
L
r
g = −GC∫
iˆ
= −
length of the stick to obtain: ( x − x)
=
0
0
ˆ
xdx
2
iˆ
Gλ
dx
0
2
iˆ
2GM
⎡ ⎛ x0
⎞ ⎛ L ⎞⎤
ln
iˆ
2 ⎢ ⎜
⎟ −
⎜
⎟
⎟⎥
L ⎣ ⎝ x0
− L ⎠ ⎝ x0
− L ⎠⎦
The Gravitational Field ( g r ) due to Spherical Objects
71 •• Two widely separated solid spheres, S1 and S2, each have radius R and
mass M. Sphere S1 is uniform, whereas the density of sphere S2 is given by
ρ(r) = C/r, where r is the distance from its center. If the gravitational field
strength at the surface of S1 is g1, what is the gravitational field strength at the
surface of S2?
Picture the Problem The gravitational field strength at the surface of a sphere is
2
given by g = GM R , where R is the radius of the sphere and M is its mass.
Express the gravitational field
strength on the surface of S1:
Express the gravitational field
strength on the surface of S2:
GM
g 1 = 2
R
GM
g 2 = 2
R
Divide the second of these equations GM
by the first and simplify to obtain: g 2
2 = R
g GM
1
2
R
= 1
g = g
⇒ 1 2
75 •• Suppose you are standing on a spring scale in an elevator that is
descending at constant speed in a mine shaft located on the equator. Model Earth
as a homogeneous sphere. (a) Show that the force on you due to Earth’s gravity
alone is proportional to your distance from the center of the planet.
(b) Assume that the mine shaft located on the equator and is vertical. Do not
neglect Earth’s rotational motion. Show that the reading on the spring scale is
proportional to your distance from the center of the planet.

Gravity
Picture the Problem There are two forces acting on you as you descend in the
elevator and are at a distance r from the center of Earth; an upward normal force
(FN) exerted by the scale, and a downward gravitational force (mg) exerted by
Earth. Because you are in equilibrium (you are descending at constant speed)
under the influence of these forces, the normal force exerted by the scale is equal
in magnitude to the gravitational force acting on you. We can use Newton’s law
of gravity to express this gravitational force.
(a) Express the force of gravity
acting on you when you are a
distance r from the center of Earth:
Using the definition of density,
express the density of Earth between
you and the center of Earth and the
density of Earth as a whole:
The density of Earth is also given by:
Equating these two expressions for ρ
and solving for M(r) yields:
Substitute for M(r) in equation (1)
and simplify to obtain:
Apply Newton’s law of gravity
to yourself at the surface of Earth
to obtain:
Substitute for g in equation (2) to
obtain:
GM ( r)
m
Fg = (1)
2
r
where M(r) is the mass of Earth
enclosed within the radius r: that is,
loser to the center of Earth than your
position.
( r)
() r
M
ρ = =
V
M
E ρ = =
4 VE
3
M
() r
= M
E
M
( r)
3
4
3 π r
M
π R
⎛ r
⎜
⎝ R
E
3
⎞
⎟
⎠
3
⎛ r ⎞
GM E ⎜ ⎟ m
R GM Em
r
=
⎝ ⎠
=
(2)
2
r R R
F g 2
3
GM Em
GM E
mg = ⇒g =
2
2
R R
where g is the magnitude of the
gravitational field at the surface of
Earth.
235
⎛ mg ⎞
Fg = ⎜ ⎟r ⎝ R ⎠
That is, the force of gravity on you is
proportional to your distance from the
center of Earth.

236
Chapter 11
(b) Apply Newton’s second law to
your body to obtain:
r
2
FN − mg = −mrω
R
2
where the net force ( − mrω
) , directed
toward the center of Earth, is the
centripetal force acting on your body.
Solving for FN yields: ⎛ mg ⎞
2
FN = ⎜ ⎟r
− mrω
Note that this equation tells us that your effective weight increases linearly with
distance from the center of Earth. However, due just to the effect of rotation, as
you approach the center the centripetal force decreases linearly and, doing so,
increases your effective weight.
77 •• A solid sphere of radius R has its center at the origin. It has a uniform
1
mass density ρ0, except that there is a spherical cavity in it of radius r = 2 R
1
centered at x = 2 R as in Figure 11-27. Find the gravitational field at points on the
x axis for x > R . Hint: The cavity may be thought of as a sphere of mass m =
(4/3)π r 3 ρ0 plus a sphere of ″negative″ mass –m.
Picture the Problem We can use the hint to find the gravitational field along the
x axis.
Using the hint, express x :
g ( )
g ( x)
= gsolid
sphere + g hollow sphere
Substitute for and g and simplify to obtain:
g
( x)
=
gsolid sphere hollow sphere
GM
=
x
solid sphere
2
⎛ 4πρ0R
G
⎜
⎝ 3
3
GM
+
1 ( x − R)
⎞⎡
1
⎟
⎟⎢
− 2
⎠⎢⎣
x 8
hollow sphere
2
2
1
⎤
( ) ⎥ ⎥ 1 2
x − 2 R ⎦
⎝
Gρ0
3 =
x
R
⎠
3 [ ]
4 3
4 1
( π R ) Gρ
− π ( R)
2
+
0
3
2
2
1 ( x − R)
81 ••• A small diameter hole is drilled into the sphere of Problem 80 toward
the center of the sphere to a depth of 2.0 m below the sphere’s surface. A small
mass is dropped from the surface into the hole. Determine the speed of the small
mass as it strikes the bottom of the hole.
Picture the Problem We can use conservation of energy to relate the work done
by the gravitational field to the speed of the small object as it strikes the bottom of
the hole. Because we’re given the mass of the sphere, we can find C by
2

Gravity
expressing the mass of the sphere in terms of C. We can then use the definition of
the gravitational field to find the gravitational field of the sphere inside its surface.
The work done by the field equals the negative of the change in the potential
energy of the system as the small object falls in the hole.
Use conservation of energy to relate
the work done by the gravitational
field to the speed of the small object
as it strikes the bottom of the hole:
Express the mass of a differential
element of the sphere:
Integrate to express the mass of
K f − Ki
+ ΔU
= 0
or, because Ki = 0 and W = −ΔU,
1 2 2W
W = mv 2 ⇒ v = (1)
m
where v is the speed with which the
object strikes the bottom of the hole
and W is the work done by the
gravitational field.
dm = ρ dV = ρ
2 ( 4π r dr)
2
M 4 π C rdr = ( 50m
) π C
the sphere in terms of C:
Solving for C yields:
Substitute numerical values and
5.
0
= ∫
0
m
M
C = 2 ( 50m
)π
1.
0×
10
C =
evaluate C: 2 ( 50m
)
11
kg
8
= 6.
37×
10 kg/m
π
Use its definition to express the gravitational field of the sphere at a distance from
its center less than its radius:
2
4π
r ρ dr
Fg
GM 0
g = = = G = G
2
2
m r r
Express the work done on the small
object by the gravitational force
acting on it:
Substitute in equation (1) and
simplify to obtain:
r
∫
r
∫
0
4π
r
r
2
2
r
C
dr 4π
C r dr
r ∫
0
= G = 2π
GC
2
r
3.0m
W = − ∫ mgdr = ( 2m)mg
v =
=
5.
0
2
m
( 2.
0m)
m(
2π
GC)
m
( 8.
0m)
π GC
2
237

238
Chapter 11
Substitute numerical values and evaluate v:
v =
−11
2 2
8 2
( 8.
0m)
π ( 6.
673×
10 N ⋅ m /kg )( 6.
37 × 10 kg/m ) = 1.
0m/s
83 ••• Two identical spherical cavities are made in a lead sphere of radius R.
The cavities each have a radius R/2. They touch the outside surface of the sphere
and its center as in Figure 11-28. The mass of a solid uniform lead sphere of
radius R is M. Find the force of attraction on a point particle of mass m located at
a distance d from the center of the lead sphere.
Picture the Problem The force of attraction of the small sphere of mass m to the
lead sphere of mass M is the sum of the forces due to the solid sphere ( FS ) and
the cavities ( ) of negative mass.
r
F r
C
Express the force of attraction: S C F F F
r r r
= +
(1)
Use the law of gravity to express the
force due to the solid sphere:
Express the magnitude of the force
acting on the small sphere due to one
cavity:
Relate the negative mass of a cavity
to the mass of the sphere before
hollowing:
Letting θ be the angle between the x
axis and the line joining the center of
the small sphere to the center of
either cavity, use the law of gravity
to express the force due to the two
cavities:
Use the figure to express cosθ :
r
F
S
GMm
= − iˆ
2
d
C
2
2
2 ⎟ GM'm
F =
⎛ R ⎞
d + ⎜
⎝ ⎠
where M′ is the negative mass of a
cavity.
3 ⎡ R 4 ⎛ ⎞ ⎤
M' = −ρ
V = −ρ
⎢ 3 π ⎜ ⎟ ⎥
⎢⎣
⎝ 2 ⎠ ⎥⎦
= −
1
8
4 3 1
( πρ R ) = − M
3
r GMm
F 2
cos iˆ
C =
θ
2 ⎛ 2 R ⎞
8 ⎜
⎜d
+
4 ⎟
⎝ ⎠
because, by symmetry, the y
components add to zero.
cosθ
=
d
2
d
R
+
4
2
8

Substitute for cosθ and simplify to
obtain:
Substitute in equation (1) and
simplify:
General Problems
r
F
C
GMm
=
2 ⎛ 2 R ⎞
4 ⎜
⎜d
+
4 ⎟
⎝ ⎠
GMmd
=
2 ⎛ 2 R ⎞
4 ⎜
⎜d
+
4 ⎟
⎝ ⎠
3 / 2
d
2
iˆ
d
R
+
4
r GMm
F iˆ
GMmd
= − + 2
d
2 ⎛ 2 R ⎞
4 ⎜
⎜d
+
4 ⎟
⎝ ⎠
=
⎡
⎢
GMm ⎢
− 1 2 ⎢ −
d
⎢ ⎧
⎢
⎨d
⎣ ⎩
2
Gravity
2
iˆ
3 / 2
3
d
4
2
R ⎫
+
4
89 •• A neutron star is a highly condensed remnant of a massive star in the
last phase of its evolution. It is composed of neutrons (hence the name) because
the star’s gravitational force causes electrons and protons to ″coalesce″ into the
neutrons. Suppose at the end of its current phase, the Sun collapsed into a neutron
star (it can’t in actuality because it does not have enough mass) of radius 12.0 km,
without losing any mass in the process. (a) Calculate the ratio of the gravitational
acceleration at the surface of the Sun following its collapse compared to its value
at the surface of the Sun today. (b) Calculate the ratio of the escape speed from
the surface of the neutron-Sun to its value today.
Picture the Problem We can apply Newton’s second law and the law of gravity
to an object of mass m at the surface of the Sun and the neutron-Sun to find the
ratio of the gravitational accelerations at their surfaces. Similarly, we can express
the ratio of the corresponding expressions for the escape speeds from the two suns
to determine their ratio.
(a) Express the gravitational force
acting on an object of mass m at the
surface of the Sun:
Solving for ag yields:
GM
F g = mag
=
R
Sun
2
Sun
Sun
2
Sun
m
⎬
⎭
iˆ
3 / 2
⎤
⎥
⎥
iˆ
⎥
⎥
⎥
⎦
GM
a g = (1)
R
239

240
Chapter 11
The gravitational force acting on an
object of mass m at the surface of a
neutron-Sun is:
Solving for a'
yields:
g
Divide equation (2) by equation (1)
to obtain:
Because M = M :
neutron-Sun
Sun
Substitute numerical values and
evaluate the ratio a :
'g ag
(b) The escape speed from the
neutron-Sun is given by:
The escape speed from the Sun is
given by:
Dividing the first of these equations
by the second and simplifying yields:
GM
F g = ma'g
=
R
neutron-Sun
2
neutron-Sun
neutron-Sun
2
neutron-Sun
m
GM
a 'g
= (2)
R
a'
a
g
g
a 'g
=
a
g
a'
a
g
g
v ' =
e
e
v =
v 'e
=
v
e
GM
R
=
GM
R
R
neutron-Sun
2
neutron-Sun
R
Sun
2
Sun
2
Sun
2
neutron-Sun
8 ⎛ 6.
96×
10 m ⎞
= ⎜
3
12.
0 10 m
⎟
⎝ × ⎠
GM
R
GM
R
R
neutron-Sun
neutron-Sun
Sun
Sun
R
Sun
neutron-Sun
M
R
=
M
R
2
neutron-Sun
2
neutron-Sun
Substitute numerical values and
8
v'e
6.
96×
10 m
evaluate v 'e ve
:
=
= 241
3
v 12.
0×
10 m
e
=
Sun
2
Sun
9
3.
36×
10
95 •• Uranus, the seventh planet in the Solar System, was first observed in
1781 by William Herschel. Its orbit was then analyzed in terms of Kepler’s Laws.
By the 1840s, observations of Uranus clearly indicated that its true orbit was
different from the Keplerian calculation by an amount that could not be accounted
for by observational uncertainty. The conclusion was that there must be another
influence other than the Sun and the known planets lying inside Uranus’s orbit.
This influence was hypothesized to be due to an eighth planet, whose predicted
orbit was described independently in 1845 by two astronomers: John Adams (no
relation to the former president of the United States) and Urbain LeVerrier. In
September of 1846, John Galle, searching in the sky at the place predicted by

Gravity
Adams and LeVerrier, made the first observation of Neptune. Uranus and
Neptune are in orbit about the Sun with periods of 84.0 and 164.8 years,
respectively. To see the effect that Neptune had on Uranus, determine the ratio of
the gravitational force between Neptune and Uranus to that between Uranus and
the Sun, when Neptune and Uranus are at their closest approach to one another
(i.e. when aligned with the Sun). The masses of the Sun, Uranus, and Neptune are
333,000, 14.5 and 17.1 times that of Earth, respectively.
Picture the Problem We can use the law of gravity and Kepler’s third law to
express the ratio of the gravitational force between Neptune and Uranus to that
between Uranus and the Sun, when Neptune and Uranus are at their closest
approach to one another.
The ratio of the gravitational force
between Neptune and Uranus to that
between Uranus and the Sun, when
Neptune and Uranus are at their
closest approach to one another is
given by:
Applying Kepler’s third law to
Uranus yields:
Applying Kepler’s third law to
Neptune yields:
Divide equation (3) by equation (2)
to obtain:
Substitute for rN in equation (1) to
obtain:
Simplifying this expression yields:
F
F
g, N-U
g, U-S
2
U
GM
M
241
( )
( ) 2
N U
2
2
rN
− rU
M NrU
= =
(1)
GM UM
S M S rN
− rU
2
rU
Cr T = (2)
2
N
3
U
Cr T = (3)
T
T
2
N
2
U
F
3
N
3 3
CrN
rN
N
= = ⇒
3 3 N U
CrU
r
⎟
U
U
⎟
⎛ T ⎞
r = r
⎜
⎝ T ⎠
g, N-U
F
F
g, U-S
g, N-U
F
g, U-S
M
=
⎛
⎜ ⎛ T
M S rU
⎜ ⎜
⎝ ⎝ T
2
N U
N
U
M N
=
⎛
⎜⎛
T ⎞ N M S⎜
⎜
⎟
U ⎝⎝
T ⎠
r
⎞
⎟
⎠
2 3
2 3
− r
⎞
−1⎟
⎟
⎠
2
U
⎞
⎟
⎟
⎠
2
2 3

242
Chapter 11
Because MN = 17.1ME and MS = 333,000ME:
F
F
g, N-U
g, U-S
=
5
3.
33×
10 M
17.
1M
E
E
⎛
⎜⎛
T
⎜ ⎜
⎝⎝
T
N
U
⎞
⎟
⎠
2 3
Substitute numerical values and evaluate
F
g, N-U
F
g, U-S
⎞
−1⎟
⎟
⎠
F
2
g, N-U
F
g, U-S
17.
1
=
⎛
5 N
3.
33 10 ⎜⎛
T ⎞
×
⎜ ⎜
⎟
U ⎝⎝
T ⎠
17. 1
−4
=
≈ 2×
10
2 3
2
⎛
5 164.
8 y ⎞
3.
33 10 ⎜⎛
⎞
× ⎜ ⎟ −1⎟
⎜ 84.
0 y ⎟
⎝⎝
⎠ ⎠
:
2 3
⎞
−1⎟
⎟
⎠
Because this ratio is so small, during the time at which Neptune is closest to
Uranus, the force exerted on Uranus by Neptune is much less than the force
exerted on Uranus by the Sun.
97 •• [SSM] Four identical planets are arranged in a square as shown in
Figure 11-29. If the mass of each planet is M and the edge length of the square is
a, what must be their speed if they are to orbit their common center under the
influence of their mutual attraction?
Picture the Problem Note that, due to the symmetrical arrangement of the
planets, each experiences the same centripetal force. We can apply Newton’s
second law to any of the four planets to relate its orbital speed to this net
(centripetal) force acting on it.
M a
M
∑
r
F
F
r
= ma
1
M
Applying radial radial
the planets gives:
θ
θ
F
2
F
1
to one of
c
1
a
M
F = 2F cosθ
+ F
2
2

Substituting for F1, F2, and θ and
2
2GM
GM
F = cos45°
+
simplifying yields: c 2
a
( 2a)
Because
2GM
= 2
a
GM
= 2
a
2
2
⎛
⎜
⎝
1 GM
+
2 2a
1 ⎞
2 + ⎟
2 ⎠
2
2
Mv 2Mv
Fc
a 2 a
= = 2
2
2Mv
GM ⎛ 1 ⎞
= ⎜ 2 +
2 ⎟
a a ⎝ 2 ⎠
Solve for v to obtain:
GM ⎛ 1 ⎞
v = ⎜1
+ ⎟ = 1.
16
a ⎝ 2 2 ⎠
2
2
Gravity
2
2
GM
a
103 ••• In this problem you are to find the gravitational potential energy of the
thin rod in Example 11-8 and a point particle of mass m0 that is on the x axis at x
= x0. (a) Show that the potential energy shared by an element of the rod of mass
1
dm (shown in Figure 11-14) and the point particle of mass m0 located at x0 ≥ 2 L
is given by
Gm0dm
GMm0
dU = − = dxs
x0
− xs
L(
x0
− xs
)
where U = 0 at x0 = ∞. (b) Integrate your result for Part (a) over the length of the
rod to find the total potential energy for the system. Generalize your function
U(x0) to any place on the x axis in the region x > L/2 by replacing x0 by a general
coordinate x and write it as U(x). (c) Compute the force on m0 at a general point x
using Fx = –dU/dx and compare your result with m0g, where g is the field at x0
calculated in Example 11-8.
Picture the Problem Let U = 0 at x = ∞. The potential energy of an element of
the stick dm and the point mass m0 is given by the definition of gravitational
potential energy: = −Gm
dm r where r is the separation of dm and m0.
dU 0
(a) Express the potential energy of
Gm0dm
dU = −
x − x
the masses m0 and dm: 0 s
The mass dm is proportional to the
size of the element dxs
:
Substitute for dm and λ to express
dm = λ dxs
M
where λ = .
L
Gm0λ
dx
dU = −
dU in terms of xs: x − x L(
x − x )
0
s
s
=
GMm
−
0
0
dx
s
s
243

244
Chapter 11
(b) Integrate dU to find the total potential energy of the system:
L /
GMm0
U = −
L ∫
=
GMm
−
L
dxs
x − x
−L
/ 2 0 s
0
2
⎛ x
ln
⎜
⎝ x
0
0
+
−
GMm
=
L
L 2 ⎞
⎟
L 2 ⎠
0
⎡ ⎛
⎢ln⎜
x
⎣ ⎝
(c) Because x0 is a general point along the x axis:
F
( x )
0
0
L ⎞ ⎛
− ⎟ − ln⎜
x
2 ⎠ ⎝
⎡
⎤
dU GMm ⎢
⎥
0 1 1
= − = ⎢ − ⎥
dx0
L ⎢
L L
x + − ⎥
0 x0
⎣ 2 2 ⎦
Further simplification yields:
F(
x )
0
Gmm
= − 2
x − L
0
0
2
L ⎞⎤
+ ⎟
2
⎥
⎠⎦
This answer and the answer given in Example 11-8 are the same.
4