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The figure shows a smooth track, a part of which is a circle of radius ‘R’. A block of mass ‘m’ is pushed against a spring of spring constant K fixed at the left end and is then released. Find the initial compression of the spring so that the block presses the track with a force ‘mg’ when it reaches the point P, where the radius of the track is horizontal

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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

Tension at horizontal position given

T_{H} = nmg

of velocity at P = 0

Velocity at bottom $V_{Q}$

$T_{P} = \frac{{mV}^{2} p}{R} = nmR$

According to LC of energy = ${TE}_{P} = {TE}_{Q}$

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

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

$\frac{1}{2} {mV}_{Q}^{2} = \frac{1}{2} mgR (n + 2)$

By conservation of energy

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

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

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