Rate of precession caused by mountain on spherical earth

[student335]

Homework Statement

R = 4000mi
phi = 60
Me = 5.972E24 kg
Mm = 5.972E16 kg
distance of procession = 100 mi

Homework Equations

I know the answer is supposed to be ~1010 years.

I also know what I am trying to do is have that e3 axis be pointing straight up, then when the mountain is placed on it, it "tips" down. So when the Earth continues spinning it know creates that little cone shape above the north pole. It will travel around this cone shape at a constant angular velocity. So when I determine that angular velocity I just divide the 100 miles by that distance. My question is how to find that angular velocity. I tried a few things, but I'm hopelessly stuck.

The Attempt at a Solution

 

[TSny]

Hello and welcome to PF!

"Free precession" usually means precession in the absence of external torques. Are you sure you aren't supposed to treat the entire system (including the mountain) using equations for free precession.

The problem refers to section 10.8 of the text. Can you find any relevant information there? For example, have you covered Euler's equations of motion for a rotating rigid body?

It might help if you tell us what type and level of course this is from.

 

[student335]

Hello and welcome to PF!

"Free precession" usually means precession in the absence of external torques. Are you sure you aren't supposed to treat the entire system (including the mountain) using equations for free precession.

The problem refers to section 10.8 of the text. Can you find any relevant information there? For example, have you covered Euler's equations of motion for a rotating rigid body?

It might help if you tell us what type and level of course this is from.

This is intermediate mechanics and yes we have "covered" up to section 10.8.

My professor just has us read the book and doesn't cover it in lecture, so I haven't gotten much help on this problem.

So I do understand Euler's equations, I just haven't read about them yet.I assume that the torque_1 and torque_2 values are zero and their respective change in angular velocity is also zero, so we only have to work with torque_3.

Torque_3 is calculated by R x F which is |R|F|sin(theta) which is RMgsin(theta)

 

[TSny]

The earth-plus-mountain object has no external torques applied to it. That's good news! You'll be able to use the Euler equations with no torques ("free precession").