A satellite of mass $1000 k g$ is rotating around the earth in a circular orbit of radius $3 R$ . What extra energy should be given to this satellite, if it is to be lifted into an orbit of radius $4 R$ ?

6.4 \times 10^{9} J

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4.8 \times 10^{9} J

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2.7 \times 10^{9} J

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3.9 \times 10^{9} J

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Solution

The correct option is C $2.7 \times 10^{9} J$

Total energy, $T . E = - \frac{G m M}{2 R}$
Initial total energy at $3 R$ , $T . E_{i} = - \frac{G m M}{6 R}$ ....(1)
Final total energy at $4 R$ , $T . E_{f} = - \frac{G m M}{8 R}$ ........(2)
$Δ E = T . E_{f} - T . E_{i} = - \frac{G m M}{8 R} + \frac{G m M}{6 R}$
$Δ E = \frac{G M m}{24 R} = \frac{g R}{24}$
Substituting the values,
$g = 10 m s^{- 2}, R = 6400 k m, m = 1000 k g$
$Δ E = 2.7 \times 10^{9} J$

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A satellite of mass 1000 kg is rotating around the earth in a circular orbit of radius 3R. What extra energy should be given to this satellite, if it is to be lifted into an orbit of radius 4R?

The correct option is C 2.7×109 J Total energy, T.E=−GmM2R Initial total energy at 3R, T.Ei =−GmM6R....(1) Final total energy at 4R, T.Ef =−GmM8R........(2) ΔE=T.Ef−T.Ei=−GmM8R+GmM6R ΔE=GMm24R=gR24 Substituting the values, g=10 ms−2,R=6400 km,m=1000 kg ΔE=2.7×109J

A satellite of mass $1000 k g$ is rotating around the earth in a circular orbit of radius $3 R$ . What extra energy should be given to this satellite, if it is to be lifted into an orbit of radius $4 R$ ?