An object is raised from the surface of the Earth (radius R ) to a height h₁ = R , requiring work W₁ . The same object is raised from the surface of the Earth to a height h₂ = 2R , requiring work W₂ . The ratio W₁ : W₂ is

Step-by-step solution

The work done in raising an object from the surface of the Earth to a height h is given by the change in gravitational potential energy: W = U = - GMm R+h - (- GMm R ) = GMmh R(R+h) For h₁ = R : W₁ = GMm(R) R(R+R) = GMm 2R For h₂ = 2R : W₂ = GMm(2R) R(R+2R) = 2GMm 3R The ratio of the work done is: W₁ W₂ = GMm 2R 2GMm 3R = 1 2 3 2 = 3 4 Thus, the ratio W₁ : W₂ is 3 : 4 . Answer: 3 : 4