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A small moon of radius r and mass m is orbiting around a planet of mass M in a circular orbit of radius R (R >> r) always keeping the same face towards the planet. If an object on the moon close to the planet is in weightlessness, then radius of the circular orbit is $R = n r \sqrt{\frac{M}{m}}$ Find n.

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answer is 1.73.

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Detailed Solution

Orbital angular speed must be equal to angular speed of revolution of moon.
$m^{1} \to$ particle mass
Question Image

$\frac{G M m^{1}}{{(R - r)}^{2}} - \frac{G m^{1} m}{r^{2}} - N = m^{1} [w^{2} R - w^{2} r]$

Similarly $m^{1}$ is in state of weightlessness then

$N = 0; \frac{G M}{{(R - r)}^{2}} - \frac{G M}{r^{2}} = w^{2} {(R - r)}^{2} \to (1)$

$& w^{2} = \frac{G M}{R^{3} \to (2)}$

From (1) & (2) $[R > > r]$
$R = r \sqrt{\frac{3 M}{m}}$

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