Question

Find an expression for the frequency ff of small-angle oscillations. Express your answer in terms of some or all of the variables MMM, RRR, and free fall acceleration ggg.

Answer 1

The frequency of the small angle of oscillation is
`\[ \boxed{f = (1)/(2\pi) \sqrt{(g)/(1.4r)}} \]`

How to determine the frequency of the small angle of oscillation?

Given that the moment of inertia
`\(I\)` about the axis of the solid sphere is calculated as:

`\[ I = (1)/(2)mr^2 + mr^2 = 0.4mr^2 + mr^2 = 1.4mr^2 \]`

The balancing torque is expressed as
`\(I\alpha = mgr\theta\)`:

`\[ 1.4mr^2 \omega^2 = mgr \]`

This equation simplifies to:

`\[ \omega^2 = (g)/(1.4r) \]`

Solving for
`\(\omega\)`:

`\[ \omega = \sqrt{(g)/(1.4r)} \]`

The frequency of the oscillation is given by
`\(f = (\omega)/(2\pi)\)`:

`\[ f = (1)/(2\pi) \sqrt{(g)/(1.4r)} \]`

Hence, the frequency of the small angle of oscillation is:

`\[ \boxed{f = (1)/(2\pi) \sqrt{(g)/(1.4r)}} \]`

See image below for missing part of the question.