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The figure shows a smooth track, a part of it is a circle of radius "R". A block of mass "m" is pushed against spring of spring constant K, and released. Find initial compression of spring such that, it presses the track with "n mg" at 'p' and does not cross it

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\sqrt{\frac{(n + 2) mgR}{k}}

\sqrt{\frac{nmgR}{k}}

\sqrt{\frac{(n - 2) mgR}{k}}

\sqrt{\frac{n}{2} \frac{mgR}{k}}

answer is A.

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Detailed Solution

T at horizontal position = nmg

$(v_{P} = 0)$

At bottom $= v_{Q}; T_{P} = \frac{{mv}_{P}^{2}}{R} = nmg$

$\frac{1}{2} {mv}_{P}^{2} + mgR = \frac{1}{2} {mv}_{Q}^{2}$

$v_{P}^{2} + 2 gR = v_{Q}^{2}$

$ngR + 2 gR = v_{Q}^{2}$

$v_{Q}^{2} = (n + 2) gR$

$\frac{1}{2} k x^{2} = \frac{1}{2} m (n + 2) gR$

$\frac{1}{2} k x^{2} = \frac{1}{2} {mv}_{Q}^{2}$

$x = \sqrt{\frac{(n + 2) mgR}{K}}$

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