Homework Set 2
Due Date
Thursday, September 16, 1999, at the beginning of class.
Lengthening of the Terrestrial Day
Consider the EarthMoon system in its current configuration.
The Moon is rotating synchronously, the Earth is rotating
asynchronously, and the orbit of the Moon about the Earth
is circular.
To answer this question, you will need the following
piece of information:
a uniform sphere of mass M and radius R, rotating with
angular frequency , has moment of inertia
, spin angular momentum , and
spin kinetic energy .
(You may have seen this in your physics courses.)

Show that the total angular momentum L and total energy E
of the EarthMoon system are given by
where subscripts `E' and `M' denote Earth and Moon
respectively, and is the orbital angular frequency.

By explicit evaluation, show that the second term is smaller
than the first in the righthand side of each expression.
We ignore both small terms in everything that follows.
Let us start by looking at how the EarthMoon system is
currently evolving.

Ignoring both small terms, carefully differentiate the
angular momentum and energy equations with respect to time.
You should find
Physically, why is dE/dt not zero? Do you expect it to
be positive or negative?

Earth's day is currently lengthening at a rate of
per year. What is ?

What is ?
How fast is the radius of the Moon's orbit changing?
(Express your answer in cm per year.)
Is the radius increasing or decreasing?
Explain your result physically.

At what rate is energy being dissipated in the Earth
as heat?

Compare your answer to 6. with the rate at which solar energy
is absorbed by the Earth. Assume that none of the solar energy
incident on Earth is reflected. The Sun's luminosity is
.
Let us now consider the end state of the EarthMoon system,
where the Earth as well as the Moon are synchronously
rotating.

Using (1), write down the total angular momentum of the
system in its current state. Do the same for the end state.

Use your answers to calculate
how long the Earth's day will be when the end state
is reached.
How long will a month last?

What will be the radius of the Moon's orbit?

How much energy will have been dissipated as heat in the course
of reaching the end state?

What extra piece of information do we need to
determine how long it will take the EarthMoon system
to reach its end state?
Astronomy 7
Fri Sep 10 17:35:15 PDT 1999