Chapter 11 Gravity

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