The work done to increase the radius of orbit of a satellite of mass $m$ revolving around a planet of mass $M$ from orbit of radius $R$ into another orbit of radius $3 R$ is :

\frac{2 G M m}{3 R}

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\frac{G M m}{R}

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\frac{G M m}{6 R}

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\frac{G M m}{24 R}

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Solution

The correct option is A $\frac{2 G M m}{3 R}$
The gravitational force ' $F$ ' on satellite of mass 'm' revolving around the planet in the orbit of radius ' $R$ ' is
$F = - \frac{G M m}{R^{2}}$
Where
$G$ is gravitational constant
$M$ is Mass of the planet.
The work done in increasing the radius of orbit of the satellite through ' $d r$ ' is
$W = F d r$
but work done is equal to change in gravitational potential ' $- d V$ '
therefore
$- d V = F d r$
$V = - \int_{R}^{3 R} F d r$
$V = - \frac{G M m}{R^{2}} \int_{R}^{3 R} - (F d r)$
$V = G M m \int_{R}^{3 R} \frac{1}{R^{2}} d r$
$V = \frac{G M m}{R} - \frac{G M m}{3 R}$
$V = \frac{2 G M m}{3 R}$
Thus work done to increase the radius of orbit of satellite is $\frac{2 G M m}{3 R}$

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The work done to increase the radius of orbit of a satellite of mass m revolving around a planet of mass M from orbit of radius R into another orbit of radius 3R is :

The work done to increase the radius of orbit of a satellite of mass $m$ revolving around a planet of mass $M$ from orbit of radius $R$ into another orbit of radius $3 R$ is :