Question

A mass of 6 × 10²⁴ kg (equal to the mass of the earth) is to be compressed in a sphere in such a way that the escape velocity from its surface is 3 × 10⁸ m s⁻¹. What should be the radius of the sphere?

Answer 1

Mass of the sphere = 6 × 10²⁴ kg
Escape velocity = 3 × 10⁸ m/s

Escape velocity is given by

\[v_e = \frac{2GM}{R}\]

\[ \Rightarrow R = \frac{2GM}{{v_e}^2}\]

\[ = \frac{2 \times 6 . 67 \times {10}^{- 11} \times 6 \times {10}^{24}}{\left( 3 \times {10}^8 \right)^2}\]

\[ = \frac{2 \times 40 . 02 \times {10}^{13}}{9 \times {10}^{16}}\]

\[ = \frac{80 . 02}{9} \times {10}^{- 3} m\]

\[ = 8 . 89 \times {10}^{- 3} m\]

\[ = 9 \ \text { mm }\]