Problem: At the bottom of a rough ellipsoidal shell \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}=1 there is a homogeneous small ball of radius r and mass m. The coefficient of friction between the ball and shell is large enough for the ball to always be in pure rolling.
The ellipsoid shell is fixed, and the coordinate system x y z is established as shown in the figure. When t=0, the ball is located at \left(0, y_0, z_0\right), and the velocity of the center of the sphere is \left (v_0, 0,0\right). It is known that y_0 \ll a, b, c, r and \frac{v_0^2}{g} \ll a, b, c, r. Find the coordinates x(t) and y(t) of the center of the sphere at any time after that.
The ellipsoid shell now rotates around the z axis at a constant angular speed \omega_0. Find the normal frequency of the small oscillations of the ball moving around the lowest point.
Please provide some hint or approach for the 1st and 2nd part of the problem. I have understood the first part somewhat but I am not able to solve the 2nd part.
The main issue here is that how will the ball oscillate and it is somewhat difficult for me to find the normal frequency of vibrations. This is the first time I am dealing with an ellipsoid.
This problem requires rigid body motion problem solving skills. If you still haven’t covered Morin’s Chapter 8, then you should certainly do so. Still, the theory is quite hard to comprehend (even David Morin says so, look at pic below), but make sure you did memorize the derivation of every formula from this chapter and solved all the exercises and problems.
That would be even harder if you didn’t try to go through previous chapters thoroughly, because there are tones of vector algebra and calculus that were explained before the 8th chapter.
Thank you for your tips. Unfortunately, I haven’t read the 8th chapter from Morin specifically, although I am familiar with the rigid body basics and rotatory motion. I need to work on my math skills though. I think that even after reading the 8th chapter, I won’t be able to solve this problem on my own
