A body initially at rest and sliding along a frictionless track from a height (as shown in the figure) just completes a vertical circle of diameter AB = D. The height is equal to [2018]

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Solution

Q.
A body initially at rest and sliding along a frictionless track from a height $h$ (as shown in the figure) just completes a vertical circle of diameter AB = D. The height $h$ is equal to [2018]

1 $\frac{3}{2} D$

2 $D$

3 $\frac{7}{5} D$

4 $\frac{5}{4} D$

Ans.
(4)

As body is at rest initially,

i.e., speed = 0

At point $A,$ speed = $v .$

As track is frictionless, so total mechanical energy will remain constant.

$∴$ ${(T.M.E)}_{i} = {(T.M.E)}_{f}$

$0 + m g h = \frac{1}{2} m v^{2} + 0 or h = \frac{v^{2}}{2 g}$

For completing the vertical circle, $v \geq \sqrt{5 g R}$

$∴ h = \frac{5 g R}{2 g} = \frac{5}{2} R = \frac{5}{4} D$

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