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JEE MainPhysicsGravitation

Two spherical planets, P and Q , have identical escape velocities. If the mass of planet P is 8 times the mass of planet Q , the ratio of the uniform mass density of planet P to that of planet Q is

Options

  1. A1 : 64
  2. B1 : 8
  3. C8 : 1
  4. D64 : 1

Correct answer

A. 1 : 64

Step-by-step solution

The escape velocity of a planet is given by v_e = 2GM R . Since the escape velocities of planets P and Q are equal, we have: 2GM_P R_P = 2GM_Q R_Q M_P R_P = M_Q R_Q R_P R_Q = M_P M_Q = 8 The uniform mass density of a planet is = M 4 3 R^3 . The ratio of their densities is: _P _Q = ( M_P M_Q ) ( R_Q R_P )^3 _P _Q = (8) ( 1 8 )^3 = 8 512 = 1 64 Thus, the ratio is 1 : 64 . Answer: 1 : 64

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