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. $9 \, mm$

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
$\frac{1}{2} \textit{m}_{\textit{e}} v _{e ⁡}^{2} = \frac{G ⁡ M ⁡_{e ⁡} m ⁡}{R ⁡_{e ⁡}}$
$\Longrightarrow \, v_{e}=\sqrt{\frac{2 G M_{e}}{R_{e}}}$
$\Longrightarrow \, \, R=\frac{2 G M_{e}}{v_{e}^{2}}$
Given,
$G =\text{6.67}\times 10^{- 1 1}N\text{m}^{2}\text{ kg}^{\text{-2}}$
$R=\frac{2 \times 6.67 \times 10^{-11} \times 6 \times 10^{24}}{\left(3 \times 10^{8}\right)^{2}}$
$ \, \, R=8.89\times 10^{- 3}$
$ \, \, R\approx 9\times 10^{- 3}m=9mm$