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# Three planets of same density and with radii $R_1,R_2$ and $R_3$ Such that $R_1=2R_2=3R_3$, have gravitational fields on the surfaces $E_1,E_2,E_3$ and escape velocities $v_1,v_2,v_3$ respectively ,then

$\begin{array}{1 1} \frac{E_1}{E_2} =1 \\ \frac{E_1}{E_3} =3 \\ \frac{v_1}{v_2} =2 \\ \frac{v_1}{v_3} =\frac{1}{3}\end{array}$

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A)
Solution :
$E= \large\frac{GM}{R^2}$
$\qquad= \large\frac{G\bigg(4/3 \pi R^3\rho\bigg)}{R^2}$
$\qquad=> E \alpha R$
$\large\frac{E_1}{E_2}= \frac{R_1}{R_2}= \frac{2R_2}{R_2}$$=2 Hence a is not correct . \large\frac{E_1}{E_3}= \frac{3R_3}{R_3}$$=3$
So b is correct
Escape velocity $v= \sqrt { \large\frac{2GM}{R}}$
$\qquad= \sqrt {\large\frac{2G}{R} \normalsize \bigg( \frac{4}{3} \pi R^3 \rho)}$
$=> v \alpha R$
$\large\frac{v_1}{v_3} =\frac{R_1}{R_2}$$=2 \therefore c is correct \large\frac{v_1}{v_3} =\frac{R_1}{R_3}$$=3$
d is not correct .