- #1

SpringPhysics

- 107

- 0

## Homework Statement

Derive the period of a sphere oscillating in a hemispherical concave surface at small angles.

## Homework Equations

Net Torque = I x alpha

Delta E_m = 0

## The Attempt at a Solution

The thing is, tried using the conservation of mechanical energy to get an expression and isolate for omega (7/10 M r^2 omega^2 = 0). Then I differentiated in order to get omega and alpha, and substituted alpha from the net torque equation (in this case, gravity = - Mg (R - r) sin theta) with the moment of inertia of the sphere. But then it gets really messy and I can't solve for omega without a theta. The bottom line is I don't think I quite understand whether circular motion plays a role here, and whether for the kinetic energy, the velocity in translational kinetic energy is (R-r) x omega (due to circular motion) or just r x omega, and whether the omega in rotational kinetic energy is v/(R-r) or just v/r.

Thanks