Two KEAM (Engineering) 2007 Multiple Choice Questions on Gravitation

The following questions which appeared in KEAM 2007 question paper are typical and of the type repeatedly asked in various entrance examinations:

(1) The change in potential energy when a body of mass ‘m’ is raised to a height nR from earth’s surface is (R = radius of the earth)

(a) mgR(n/n–1) (b) mgR (c) mgR(n/n+1)

(d) mgR(n²/n² +1) (e) mgR/n

This question appears often in entrance question papers. There may be slight change in the wording (as for example, “what is the work done in lifting a body of mass m, from the earth’s surface, through a height nR?”).

Gravitational potential energy of a mass ‘m’ at a height ‘h’ is given by

U = – GMm/(R+h) where M is the mass of the earth and G is the Gravitational constant.

Since h = nR, the change in potential energy is

– GMm/(R+nR) – (–GMm/R) = (GMm/R)[1 – 1/(1+n)] = (GMm/R)[n/(1+n)]

Since g = GM/R², the change in potential energy becomes mgRn/(n+1), given in option (c).

(2) The escape velocity of a body on the surface of the earth is 11.2 km/s. If the mass of the earth is doubled and its radius is halved, the escape velocity becomes

(a) 5.6 km/s (b) 11.2 km/s (c) 22.4 km/s

(d) 44.8 km/s (e) 67.2 km/s

The escape velocity (v_e) of a body on the surface of the earth is given by

v_e = √(2GM/R)

Therefore we have √(2GM/R) = 11.2 km/s

If the mass (M) of the earth is doubled and its radius (R) is halved, the escape velocity becomes √(2×4GM/R) = 2×√(2GM/R) = 22.4 km/s.