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- Homework Statement
- A ball of mass m and radius r is put on a smooth surface having radius R (R > r). The ball is given a small displacement and then released so that it moves back and forth at the bottom of the surface. What is the angular frequency of the ball?

a. ##\omega = \sqrt{\frac{2g}{R}}##

b. ##\omega = \sqrt{\frac{g}{r}}##

c. ##\omega = \sqrt{\frac{g}{R}}##

d. ##\omega = \sqrt{\frac{g}{R-r}}##

e. ##\omega = \sqrt{\frac{g}{R+r}}##

- Relevant Equations
- Restoring force = m.a

Small angle approximation

##a=- \omega^{2} x##

When given a small displacement ##x##, the equation for m is:

(i) N sin θ = m.a where N is the normal force acting on the ball and θ is angle of the ball with respect to vertical.

(ii) N cos θ = m.g

So:

$$\tan \theta = \frac a g$$

$$\frac x R = \frac{\omega^{2} x}{g} \rightarrow \omega = \sqrt \frac{g}{R}$$

Is this correct? The size of m is not important?

Thanks