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JEE Main 2024 · Physics · Gravitation · hard

A satellite of mass m circles the Earth (radius R, surface gravity g) in a circular orbit of radius 3R about the Earth's centre. It is shifted to a…

Single correct option

A satellite of mass mm circles the Earth (radius RR, surface gravity gg) in a circular orbit of radius 3R3R about the Earth's centre. It is shifted to a circular orbit of radius 4R4R. The minimum work required is
  1. A. mgR12\dfrac{mgR}{12}
  2. B. mgR6\dfrac{mgR}{6}
  3. C. mgR8\dfrac{mgR}{8}
  4. D. mgR24\dfrac{mgR}{24}
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Answer: (D) mgR24\dfrac{mgR}{24}

For a circular orbit of radius rr, total energy E=−GMm2rE=-\dfrac{GMm}{2r} with GM=gR2GM=gR^2. Work =Ef−Ei=GMm2(13R−14R)=mgR22⋅112R=mgR24=E_f-E_i=\dfrac{GMm}{2}\left(\dfrac1{3R}-\dfrac1{4R}\right)=\dfrac{mgR^2}{2}\cdot\dfrac{1}{12R}=\dfrac{mgR}{24}. Using the potential-energy change alone, GMm(1/3R−1/4R)=mgR/12GMm(1/3R-1/4R)=mgR/12, ignores the change in kinetic energy and is the usual wrong option.
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