First slide

Gravitational force and laws

Question

A small body of superdense material, whose mass is twice the mass of the earth but whose size is very small compared to the size of the earth, starts from rest at a height H <<R above the earth's surface, and reaches the earth's surface in time t. Then t is equal to

Moderate

A

$\sqrt{2 H / g}$
B

$\sqrt{H / g}$
C

$\sqrt{2 H / 3 g}$
D

$\sqrt{4 H / 3 g}$

Solution

As the masses of the body and the earth are comparable, they will move towards their centre of mass, which remains stationery .
Hence the body of mass 2m move through distance H/3.
As per the relation $m_{1} r_{1} = m_{2} r_{2}$ and time to reach the earth surface
$\begin{array}{l} = \sqrt{\frac{2 h}{g}} \\ = \sqrt{\frac{2 H / 3}{g}} = \sqrt{\frac{2 H}{3 g}} \end{array}$

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