Initially a satellite of 100 kg is in a circular orbit of radius 1.5Rₑ. This satellite can be moved to a circular orbit of radius 3Rₑ by supplying α × 10⁶J of energy. The value of α i $s ___________.$
(Take Radius of Earth Rₑ = 6 × 10⁶ m and g = 10 m·s⁻²)

Question

Initially a satellite of 100 kg is in a circular orbit of radius 1.5Rₑ. This satellite can be moved to a circular orbit of radius 3Rₑ by supplying α × 10⁶J of energy. The value of α i $s ___________.$

(Take Radius of Earth Rₑ = 6 × 10⁶ m and g = 10 m·s⁻²)

G1 Physics · Accepted Answer

The total mechanical energy of a satellite in a circular orbit of radius r is E = - GMₑm ────── 2r . Initial radius: r₍i₎ = 1.5Rₑ. Final radius: r₍f₎ = 3Rₑ. Therefore, ΔE = E𝒇-Eᵢ = - GMₑm ────── 6Rₑ + GMₑm ────── 3Rₑ = GMₑm ────── 6Rₑ . At the Earth's surface, g = GMₑ ───── Rₑ² . Hence, GMₑ ───── Rₑ = gRₑ. Therefore, ΔE = mgRₑ ────── 6 . Substitute m = 100 kg, g = 10 m·s⁻², Rₑ = 6 × 10⁶ m. Thus, ΔE = 100 × 10 × 6 × 10⁶ ────────────────── 6 = 1000 × 10⁶ J. Hence, α = 1000. Correct option: D.

Initially a satellite of 100 kg is in a circular orbit of radius 1.5Rₑ. This satellite can be moved to a circular orbit of radius 3Rₑ by supplying α × 10⁶J of energy. The value of α i s ___________. (Take Radius of Earth Rₑ = 6 × 10⁶ m and g = 10 m·s⁻²)