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Q.

A small ball of mass m is released from a height R above the surface of a planet of mass M and radius R as shown in fig. There is a narrow groove inside the planet in which a spring of spring constant K and natural length R. Its one end is fixed at O. If the mass ‘m’ moves R/2 distance inside planet before coming to rest, then value of K will be $\frac{x G M m}{R^{3}}$ . Find the value of $x$ .

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Detailed Solution

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By MEC,

$\begin{array}{l} K E_{i} + P E_{i} = K E_{f} + P E_{f} \\ 0 - \frac{G M m}{2 R} = 0 - \frac{11}{8} \frac{G M m}{R} + \frac{1}{2} K {(\frac{R}{2})}^{2} \\ K = \frac{7 G M m}{R^{3}} \end{array}$

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A small ball of mass m is released from a height R above the surface of a planet of mass M and radius R as shown in fig. There is a narrow groove inside the planet in which a spring of spring constant K and natural length R. Its one end is fixed at O. If the mass ‘m’ moves R/2 distance inside planet before coming to rest, then value of K will be xGMmR3 . Find the value of x.