A metal ball of mass 'm' is put at the point A of a loop track and the vertical

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A metal ball of mass 'm' is put at the point A of a loop track and the vertical distance of A from the lower most point of track is 8 times the radius 'R' of the circular part. The linear velocity of ball when it rolls of the point B to a height 'R' in the circular track will be

A Metal Ball Of Mass 'm' Is Put At The Point A Of A Loop Tra...

Options:

A . [10gR]1/2

B . 7[gR10]1/2

C . [7gR5]1/2

D . [5gR]1/2

Answer: Option A
:
A
Applying the conservation of energy at points A and B, we have

m g (8 R) = \frac{m v^{2}}{2} + \frac{1}{2} I ω^{2} + m g R

or

m g (8 R) = \frac{m v^{2}}{2} + \frac{1}{2} (\frac{2}{5} m R^{2}) {(\frac{v}{R})}^{2} + m g R

= \frac{7}{10} m v^{2} + m g R

m g (8 R - R) = \frac{7}{10} m v^{2}

[10 g R]^{1 / 2}

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