837
Chapter 11
Gravity
Conceptual Problems
*1
•
(
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.
2
•
Determine the Concept
We can apply Newton’s 2
nd
law and the law of gravity to the
satellite to obtain an expression for its speed as a function of the radius of its orbit.
Apply Newton’s 2
nd
law to the
satellite to obtain:
∑
=
=
r
v
m
r
GMm
F
2
2
radial
where
M
is the mass of the object the
satellite is orbiting and
m
is the mass of the
satellite.
Solve for
v
to obtain:
r
GM
v
=
Thus the speed of the satellite is
independent of its mass and:
correct.
is
)
(
c
3
••
Picture the Problem
The acceleration due to gravity varies inversely with the square of
the distance from the center of the moon.
Express the dependence of the
acceleration due to the gravity of the
moon on the distance from its
center:
2
1
r
a'
∝
Express the dependence of the
acceleration due to the gravity of the
moon at its surface on its radius:
2
M
1
R
a
∝

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Chapter 11
838
Divide the first of these expressions
by the second to obtain:
2
2
M
r
R
a
a'
=
Solve for
a
′
:
(
)
a
a
R
R
a
r
R
a'
16
1
2
M
2
M
2
2
M
4
=
=
=
and
correct.
is
)
(
d
4
•
Determine the Concept
Measurement of
G
is difficult because masses accessible in the
laboratory are very small compared to the mass of the earth.
5
•
Determine the Concept
The escape speed for a planet is given by
R
Gm
v
2
e
=
.
Between
v
e
depends on the square root of
M
, doubling
M
increases the escape speed by a
factor of
2
and
correct.
is
)
(
a
6
••
Determine the Concept
We can take careful measurements of its position in order to
determine whether its trajectory is an ellipse, a hyperbola, or a parabola. If the path is an
ellipse, it will return; if its path is hyperbolic or parabolic, it will not return.
7
••
Determine the Concept
The gravitational field is proportional to the mass within the
sphere of radius
r
and
inversely proportional to the square of
r
, i.e., proportional
to
.
2
3
r
r
r
=
*8
•
Determine the Concept
Let
m
represent the mass of Mercury,
M
S
the mass of the sun,
v
the orbital speed of Mercury, and
R
the mean orbital radius of Mercury. We can use
Newton’s 2
nd
law of motion to relate the gravitational force acting on the Mercury to its
orbital speed.
Use Newton’s 2
nd
law to relate the
gravitational force acting on
Mercury to its orbital speed:
R
v
m
R
m
GM
F
2
2
S
net
=
=
Simplify to obtain:
U
R
m
GM
R
m
GM
mv
2
1
S
2
1
S
2
1
2
2
1
−
=
⎟
⎠
⎞
⎜
⎝
⎛
−
−
=
=