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Question:

A planet of mass M has two natural satellites with masses m1 and m2. The radii of their circular orbits are R1 and R2 respectively. Ignore the gravitational force between the satellites. Define v1, L1, K1 and T1 to be respectively, the orbital speed, angular momentum, kinetic energy and time period of revolution of satellite 1; and v2, L2, K2 and T2 to be the corresponding quantities of satellite 2. Given m1/m2 = 2 and R1/R2 = 1/4, match the ratios in List - I to the numbers in List - II.
List - I
List - II
P. v1/v2

  1. 1/8
    Q. L1/L2
  2. 1
    R. K1/K2
  3. 2
    S. T1/T2
  4. 8

P→2; Q→3; R→1; S→4

P→4; Q→2; R→1; S→3

P→3; Q→2; R→4; S→1

P→2; Q→3; R→4; S→1

Solution:

The correct option is B
P→3; Q→2; R→4; S→1
GMm1/R1² = m1v1²/R1
v1² = GM/R1, v2² = GM/R2
v1/v2 = √(R2/R1) = √4 = 2 (P)
L = mvr
L1/L2 = (m1v1R1)/(m2v2R2) = (2)(2)(1/4) = 1 (Q)
K = 1/2mv²
K1/K2 = (m1v1²)/(m2v2²) = 2(2)² = 8 (R)
T = 2πR/v
T1/T2 = (R1/v1)/(R2/v2) = (R1/R2)(v2/v1) = (1/4)(1/2) = 1/8 (S)