Conservation of Angular momentum

[CalcYouLater]

Homework Statement

The moon will gradually slow down the Earth's rotation period. If the total angular momentum is conserved in an Earth-moon system, how long will the orbital period of the moon be when both bodies will keep same face toward each other? How far will the two bodies be from each other?

Hint: Use Kepler's 3rd law and a spreadsheet to find numerical solutions

Homework Equations

$$L=m{\omega}r^2+I_e\omega_e$$

omega is the orbital angular velocity of the moon and omega_e is the angular velocity of the earth.

$$\tau^2=(\frac{4\pi^2}{G(M_e+M_m)}r^3)$$

Where tau is the period, M_m is the mass of the moon, and M_e is the mass of the Earth.

The Attempt at a Solution

This is the last of a multiple part problem. I didn't have a problem with the other parts, but I'm not sure what to do here. I know that the final angular momentum will equal the current one, but from here it looks like I have 1 equation and 2 unknowns (omega final and r final). The phrase "use a spreadsheet" makes me think that it is going to end up being a differential equation that I can solve using Euler's numerical methods in excel or something like excel. Any ideas would be appreciated.

 

[anathema]

Hello, forum user!

First of all, great job on recognizing that the final angular momentum will be equal to the current one in a conserved system. You are correct in thinking that you have 1 equation and 2 unknowns, but this can be solved using numerical methods.

To start, we can use Kepler's 3rd law to relate the orbital period (tau) and the distance between the Earth and the moon (r). We can rearrange the equation to solve for r:

r = (G(M_e+M_m)/4pi^2)^(1/3) * tau^(2/3)

Next, we can use the equation for angular momentum to relate the angular velocities of the Earth and the moon:

m{\omega}r^2+I_e\omega_e = constant

Since we are looking for the final state where both bodies keep the same face towards each other, we can assume that the final angular velocities are equal (omega = omega_e). This gives us:

m{\omega}r^2+I_e\omega = constant

We can substitute in the expression for r from Kepler's 3rd law:

m{\omega}[(G(M_e+M_m)/4pi^2)^(1/3) * tau^(2/3)]^2+I_e\omega = constant

Now, we have an equation with only one unknown (tau), which we can solve using a spreadsheet. You can set up a table with different values of tau and calculate the corresponding values of r. Then, you can plot the values and find the point where the distance between the Earth and the moon is equal to the radius of the moon (since both bodies will keep the same face towards each other, the distance between them will be equal to the radius of the moon).

Once you have found the corresponding value of tau, you can use the equation for r to calculate the distance between the Earth and the moon at that time. This will give you the final orbital period and distance between the Earth and the moon when both bodies keep the same face towards each other.

I hope this helps! Let me know if you have any further questions. Good luck with your calculations!