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Friction and Vertical Circular Motion Problems

A water bucke...

Question

A water bucket of mass ' $m$ ' is revolved in a vertical circle with the help of a rope of length ' $r$ '. If the velocity of the bucket at the lowest point is $\sqrt{7 g r}$ . Then the velocity and tension in the rope at the highest point are:

A

\sqrt{3 g r}, 2 m g

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B

\sqrt{2 g r}, m g

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C

\sqrt{g r}, m g

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D

Z e r o

,

Z e r o

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Solution

The correct option is A $\sqrt{3 g r}, 2 m g$
$V_{b} = \sqrt{7 g r}$
Work Energy Theorem applied between top and bottom.
$m g (2 R) = \frac{1}{2} m (V_{b})^{2} - \frac{1}{2} m (V_{T})^{2}$
$2 m g r = \frac{7}{2} m g r - \frac{1}{2} m (V_{T})^{2}$
$V_{T} = \sqrt{3 g r}$

At top point:
$T = \frac{m V_{T}^{2}}{r} - m g$
$= m \frac{3 g r}{r} - m g$
$= 2 m g$

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