Question

On the surface of earth, the gravitational field is $E_g$ and gravitational potential is $V(R =$ radius of earth).

Column I		Column II
i	At a height $h = R$, value of $E_g$	p	decreases by a factor $\frac{1}{4}$
ii	At a depth $d = \frac{R}{2}$, value of $E_g$	q	decreases by a factor $\frac{1}{2}$
iii	At a height $h = R$, value of $V$	r	increases by a factor $\frac{11}{8}$
iv	At a depth $d = \frac{R}{2}$, value of $V$	s	increases by a factor $2$
		t	decreases by a factor $\frac{11}{8}$

Match the given columns and select the correct option from the codes given below.

• A. (i)—(s), (ii)—(r), (iii)—(t), (iv)—(p)
• B. (i)—(p), (ii)—(q), (iii)—(r), (iv)—(s)
• C. (i)—(p), (ii)—(q), (iii)—(s), (iv)—(t)
• D. (i)—(r), (ii)—(s), (iii)—(p), (iv)—(q)

Answer 1

Answer: (C)

$E_h = \frac{GM}{(R+h)^2} = \frac{GM}{4R^2} = \frac{E}{4}$
$V_h = -\frac{GM}{2R} = -\frac{V}{2}$
$E_d = E (1 - \frac{d}{R}) = \frac{E}{2}$
$V_d = -\frac{GM}{2R^3} (3R^2 - r^2)$
$ = -\frac{GM}{2R^3}(3R^2 - \frac{R^2}{4})$
$ = -\frac{11GM}{8R}$