Phys 341: Homework #5 Solutions
1.
(a) If the Earth and Moon are tidally locked and the Earth’s rotation period is 47 days, the
the Moon’s orbital period is also
P
= 47 days = 4
.
1
×
10
6
s. Then from Kepler’s Third
Law, the semimajor axis of the Moon’s orbit is
a
=
GM
⊕
P
2
4
π
2
1
/
3
=
(6
.
67
×
10

8
g

1
cm
3
s

2
)
×
(5
.
974
×
10
27
g)
×
(4
.
1
×
10
6
s)
2
4
π
2
1
/
3
=
5
.
5
×
10
10
cm
=
550
,
000 km
For comparison, the Moon’s current semimajor axis is about 384,000 km.
(b) The Earth’s rotation period is increasing by about
dP/dt
= 0
.
0016 seconds per century.
If the rate of change remains constant, the amount of time it will take for the period to
increase from the current value of
P
1
= 1 day to the final value of
P
2
= 47 days is
T
∼
P
2

P
1
dP/dt
∼
(47

1)
×
86
,
400 s
0
.
0016 s/century
∼
2
.
5
×
10
9
centuries
∼
2
.
5
×
10
11
yr
A different approach is to say that the Moon’s semimajor axis is increasing at a rate of
da/dt
∼
3 cm/yr. If that rate of change remains constant, then the amount of time it
will take for the semimajor axis to increase from the current value of
a
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 Spring '08
 Keeton
 Physics, Work, Tidal force, semimajor axis, Phobos

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