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A particle of mass ‘M’ is released from rest at one end of the smooth tunnel dug along a diameter of a uniform solid sphere of man 2M and radius R as shown. Find the speed of the particle w.r.t to tunnel when the particle reaches the centre of the sphere

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a

$\sqrt{\frac{4 G M}{R}}$

b

$\sqrt{\frac{3 G M}{R}}$

c

$\sqrt{\frac{G M}{R}}$

d

$3 \sqrt{\frac{G M}{R}}$

answer is B.

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Detailed Solution

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$v_{2} \to$ speed of sphere
Question Image
$v_{1} \to$ speed of particle
From linear momentum conservation
$O = 2 M v_{2} - M v_{1} ........ (1)$
Energy conservation
$- \frac{2 G M^{2}}{R} = - \frac{3 G M^{2}}{R} + \frac{1}{2} M V_{1}^{2} + \frac{1}{2} 2 M V_{2}^{2} ....... (2)$

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A particle of mass ‘M’ is released from rest at one end of the smooth tunnel dug along a diameter of a uniform solid sphere of man 2M and radius R as shown. Find the speed of the particle w.r.t to tunnel when the particle reaches the centre of the sphere