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A hole is drilled along the diameter of the earth and pen is dropped into it. The time taken by it is reach other end of the earth is

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42 min

answer is B.

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Detailed Solution

At a radial distance r (r<R), acceleration due to gravity is given by $\large g\, = \,{g_s}.\frac{r}{R}$

$\large \therefore \,m.\frac{{{d^2}r}}{{d{t^2}}} = \, - mg\, = \, - m{g_s}.\frac{r}{R}$

$\large \Rightarrow \,\frac{{{d^2}r}}{{d{t^2}}}\, + \left( {\frac{{{g_s}}}{R}} \right)r\, = \,0$

$\large \Rightarrow \,\frac{{{d^2}r}}{{d{t^2}}} + {\omega ^2}r\, = 0$

The equation represents SHM whose time period is given by $\large T\, = \,\frac{{2\pi }}{w}\, = \,2\pi \sqrt {\frac{R}{{{g_s}}}}$

$\large \therefore \,\frac{T}{2}\, = \,\pi \sqrt {\frac{R}{{{g_s}}}} = \,\pi \sqrt {\frac{{64 \times {{10}^5}}}{{9.8}}} \sec$ = 42min

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