JEE MAIN - Physics (2002 - No. 70)
Energy required to move a body of mass $$m$$ from an orbit of radius $$2R$$ to $$3R$$ is
$${{GMm} \over {12{R^2}}}$$
$${{GMm} \over {3{R^2}}}$$
$${{GMm} \over {8R}}$$
$${{GMm} \over {6R}}$$
Explanation
Gravitational potential energy E = $$ - {{GMm} \over r}$$
where M = mass of earth
m = mass of body
r = radius of earth
Energy required to move a body of mass $$m$$ from an orbit of radius $$2R$$ to $$3R$$
$$=$$ (Potential energy of the Earth-mass system when mass is at distance $$3R$$ ) $$-$$ (Potential energy of the Earth-mass system when mass is at distance $$2R$$)
$$ = {{ - GMm} \over {3R}} - \left( {{{ - GMm} \over {2R}}} \right)$$
$$ = {{ - GMm} \over {3R}} + {{GMm} \over {2R}}$$
$$ = {{ - 2GMm + 3GMm} \over {6R}} = {{GMm} \over {6R}}$$
where M = mass of earth
m = mass of body
r = radius of earth
Energy required to move a body of mass $$m$$ from an orbit of radius $$2R$$ to $$3R$$
$$=$$ (Potential energy of the Earth-mass system when mass is at distance $$3R$$ ) $$-$$ (Potential energy of the Earth-mass system when mass is at distance $$2R$$)
$$ = {{ - GMm} \over {3R}} - \left( {{{ - GMm} \over {2R}}} \right)$$
$$ = {{ - GMm} \over {3R}} + {{GMm} \over {2R}}$$
$$ = {{ - 2GMm + 3GMm} \over {6R}} = {{GMm} \over {6R}}$$
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