Chapter 11 Gravity

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