Question

Taking the earth to be a uniform sphere of radius $6400\text{km}$ and the value of g at the surface to be $10{\text{ms}}^{-2}$, the energy needed to raise a satellite of mass $2000\text{kg}$ to a height of $800\text{km}$ above the earth's surface and to set it into circular orbit at that altitude wil be

• A. $7.3\times {10}^{10}\text{J}$
• B. $5.2\times {10}^{10}\text{J}$
• C. $6.9\times {10}^{10}\text{J}$
• D. $1.4\times {10}^{10}\text{J}$

Answer 1

Answer: (D)

$\text{Initial energy}=\frac{GMm}{R}$
$\text{Here},m=\text{mass of the satellite and}M=\text{mass of the earth}$
$\text{Final nergy}=\frac{GMm}{(R+h)}$
$\text{Change in energy}=\text{Final energy-Initial energy}$
$\frac{GMm}{R+h}-(-\frac{GMm}{R})=\frac{GMm}{R}-\frac{GMm}{R+h}=GMm\left[\frac{h}{R(R+h)}\right]=\frac{GM}{R^2}m\left[\frac{Rh}{R+h}\right]$
$\text{Energy needed}=g.m\left(\frac{Rh}{R+h}\right)$
where,$g=\frac{GM}{R^2}$
$\therefore \text{Energy needed}=10\times 2000\left[\frac{6400\times {10}^3\times 800000}{6400\times {10}^3+800000}\right]=1.4\times {10}^{10}\text{J}$