Oscillation and Rotation: Can You Solve These Equilibrium and Motion Problems?

[benf.stokes]

Oscillation and Rotation

Homework Statement

Question nr 1:

You have ruler of length L and thickness 2d resting, in equilibrium , on a cylindrical body of radius r. Slightly unbalancing the ruler, and existing attrition between the surfaces prove that the ruler has a oscillatory motion of period:
$$T = 2\cdot \pi\cdot \sqrt{\frac{L^2}{12\cdot g\cdot (r-d)}}$$

Question nr 2:

Assuming the sphere roles down without sliding prove that the acceleration of it's center of mass is:

$$a= \frac{g\cdot \sin(\theta)}{1+\frac{2}{5}\cdot \frac{1}{1-\frac{1}{4}\cdot \alpha^2}}$$

Where g is the gravitational acceleration and
$$\alpha= \frac{L}{R}$$

Note: The moment of inertia of the sphere is:
$$I= \frac{2}{5}\cdot M\cdot R$$

Homework Equations

$$T=\frac{2\cdot \pi}{\omega}$$

$$\tau= F\cdot r\cdot \sin(\varphi)$$

The Attempt at a Solution

At question nr 1 I can't wrap my mind about the idea that the ruler won't immediately begin to fall and in question nr 2 I get to:
$$a= \frac{g\cdot \sin(\theta)}{1+\frac{2}{5}\cdot \sqrt{\frac{1}{1-\frac{1}{4}\cdot \alpha^2}}}$$

 

[darkxponent]

moment of inertia of sphere is I=(2/5).M.R^2