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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?
Solution
Mass of the sphere = 6 × 1024 kg
Escape velocity = 3 × 108 m/s
Escape velocity is given by
\[v_e = \frac{2GM}{R}\]
\[ \Rightarrow R = \frac{2GM}{{v_e}^2}\]
\[ = \frac{2 \times 6 . 67 \times {10}^{- 11} \times 6 \times {10}^{24}}{\left( 3 \times {10}^8 \right)^2}\]
\[ = \frac{2 \times 40 . 02 \times {10}^{13}}{9 \times {10}^{16}}\]
\[ = \frac{80 . 02}{9} \times {10}^{- 3} m\]
\[ = 8 . 89 \times {10}^{- 3} m\]
\[ = 9 \ \text { mm }\]
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