- #1
songoku
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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