Question

A mass of $6\times 10^{24}$ $kg$ is to be compressed in a sphere in such a way that the escape velocity from the sphere is $3\times 10^8$ . What should be the radius of the sphere?

• A. $9km$
• B. $9m$
• C. $9mc$
• D. $9mm$

Answer 1

Answer: (D)

Let escape velocity be $v_e$ , then kinetic energy is
$=\frac{1}{2}m{v}_e^2$
and escape energy $=+\frac{G{M}_{\mathrm{e}}m}{R_e}$
Equating Eqs. (i) and (ii), we get
$\frac{1}{2}{\text{m}}_{\text{e}}{v}_e^2=\frac{GM{}_{e}m}{R{}_e}$
$⟹v_e=\sqrt{\frac{2G{M}_e}{R_e}}$
$⟹R=\frac{2G{M}_e}{v_e^2}$
Given,
$G=\text{6.67}\times 10^{-11}N{\text{m}}^2{\text{ kg}}^{\text{-2}}$
$R=\frac{2\times 6.67\times 10^{-11}\times 6\times 10^{24}}{{\left(3\times 10^8\right)}^2}$
$R=8.89\times 10^{-3}$
$R\approx 9\times 10^{-3}m=9mm$