Question

On the surface of earth acceleration due to gravity is g and gravitational potential is V. Match the Column I with Column II and mark the correct option from the codes given below.
Column I
(A) At height h = R, value of g
(B) At depth h = $\frac{R}{2}$ , value of g
(C) At height h = R, value of V
(D) At depth h = $\frac{R}{2}$ , value of V
Column II
(p) decreases by a factor $\frac{1}{4}$
(q) decrease by a factor $\frac{1}{2}$
(r) decrease by a factor $\frac{11}{8}$
(s) increases by a factor 2
(t) None

Moderate

Solution

As $g = \frac{G M}{R^{2}}, V = - \frac{G M}{R}$
At height $h = R, g = \frac{g}{{(1 + \frac{h}{R})}^{2}} = \frac{g}{4} \Rightarrow V^{'} = - \frac{G M}{2 R}$
i.e. g decrease by a factor 1/4 and V also decreases by a factor of 1/2.
At depth $h = \frac{R}{2}$
$g^{'} = g (1 - \frac{h}{R}) = g (1 - \frac{1}{2}) = \frac{g}{2}$
$V^{'} = - \frac{G M}{R^{3}} (15 R^{2} - 0.5 r^{2}) = - \frac{G M}{R^{3}} (15 R^{2} - \frac{1}{8} R^{2}) = - \frac{11}{8} \frac{G M}{R}$
i.e. g decrease by a factor $\frac{1}{2}$ and V' also decreases by a factor $\frac{11}{8}$ .

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