Initially a satellite of 100 kg is in a circular orbit of radius 1.5R_E. This satellite can be moved to a circular orbit of radius 3R_E by supplying α× 10⁶ J of energy The value of α is ____ .
(Take Radius of Earth R_E=6× 10⁶ m and g=10 m/s² )

Solution

Physics sir · 4 steps

The energy you supply is the rise in total orbital energy, -GMm/2r, between the two orbits.

1. Energy in an orbit. A satellite in a circular orbit of radius r has total energy E = -GMm/2r. Moving it to a bigger orbit makes E less negative. The energy supplied equals that rise.

   W = E₂ - E₁ = GMm/2(1/r₁-1/r₂), GM = gR_E²

2. Simplify the bracket. With r₁ = 1.5R_E and r₂ = 3R_E, the bracket is 2/3R_E-1/3R_E = 1/3R_E. So W = (gR_E² m)/2×1/3R_E = mgR_E/6.
3. Put in numbers. W = (100 × 10 × 6×10⁶)/6 = 1×10⁹ J = 1000×10⁶ J.
4. So. α = 1000, which is option (b).

The idea to keep. To shift a satellite between circular orbits, supply the change in total energy -GMm/2r, not the change in potential energy alone. Replace GM with gR_E².

Answer from the NTA final answer key. Solution by PrepWiser.