# Rotation: the earth-moon system

1. Dec 9, 2004

### freek

The Earth has a rotational period of 23 hours, 56 minutes. The moon has a rotational period of 27.3 days and an orbital period of 27.3 days. The moon has a radius of 1.74 x10^6 m, an orbital radius of 3.82 x10^8 m, and a mass of 7.36 x10^22 kg. The earth has a radius of 6.37 x10^6 m and a mass of 5.98 x10^24 kg. Assume that the moon has a circular orbit around the earth and that the three angular angular momentum vectors are parallel (which isn't really the case).

a) calculate the angular momentum of the Earth-moon system. All of the rotations are counterclockwise.

i used the relation T=2pi/omega to solve for omega for both earth and the moon. assuming both the earth and the moon are solid spheres, I=.4MR^2. then i used L=I times omega to get L for both the earth and the moon. and then adding earth and moon's L's gives me the L for the system. right?

b) If, due to torques internal to the Earth-mmon system, the earth's day is being lengthened by the rate of 1.48 ms/century, calculate the average torque exerted on the Earth by the moon.

since T is increasing, Earth's angular velocity has to be decreasing. but how do i go about finding the torque? rFsin theta? dL/dt ?

i'm hoping someone can help me get a good start. thanks.

2. Dec 9, 2004

### Tide

There are contributions to the total angular momentum from both the rotation of each object about its own axis and rotation of both about a common axis (where the center of mass lies). You have to take both into account.

For the second problem, you can find dL/dt from the information provided:

$$\tau = I \alpha = I \frac {d \omega}{dt} = \frac {dL}{dt}$$

where $I$ presumably remains the same.

3. Dec 9, 2004

### Staff: Mentor

So far, so good. But don't forget the angular momentum of the moon orbiting the earth.

Figure out the angular acceleration, then use Newton's 2nd law: $\tau = I \alpha$.

4. Dec 9, 2004

### freek

so, theres four partial angular momentums that make up the total angular momentum of the system:
1 and 2- the earth and the moon rotating about their own axises.
3 and 4- rotation of both about a common axis that goes through the center of mass for the system (the moon orbiting around the earth).

and to find the I's for the moon and the earth rotating about that common axis, i have to use the parallel axis theorem. right? i'm just thinking out loud cos the concepts here are more abstract that what i'm used to previously. :-]

and for the second part, how would i figure out the angular acceleration? thanks for the replies.

5. Dec 9, 2004

### Tide

Angular acceleration = change in angular velocity per unit time!

You're given the rate of change of the period so you just need to relate the period to the angular velocity.

6. Dec 10, 2004

### Staff: Mentor

Tide is, of course, correct (as usual ) about the moon and earth orbiting about their common center of mass. However, you are asked to treat the simpler case of the moon having a circular orbit around the earth:
"Assume that the moon has a circular orbit around the earth and that the three angular momentum vectors are parallel (which isn't really the case)."​