Question

If the earth shrinks such that its density becomes 8 times to the present value, then the new duration of the day in hours will be

• A. 24
• B. 12
• C. 6 $ \sqrt{2} $
• D. 3

Answer 1

The Correct Option is C

If d is the density of the material of earth, then $ g=\frac{GM}{{{R}^{2}}}=\frac{G.\frac{4}{3}\pi {{R}^{3}}d}{{{R}^{2}}} $ $ =\frac{4}{3}G\pi Rd $ or $ g\propto d $ Now, $ T=2\pi \sqrt{\frac{l}{g}} $ $ \Rightarrow $ $ T=2\pi \sqrt{\frac{l}{d}} $ or $ T\propto \frac{1}{\sqrt{d}} $ $ \therefore $ $ \frac{{{T}_{1}}}{{{T}_{2}}}=\sqrt{\frac{{{d}_{2}}}{{{d}_{1}}}} $ Given, $ {{d}_{1}}=d,\,{{d}_{2}}=8d $ $ \Rightarrow $ $ \frac{{{T}_{1}}}{{{T}_{2}}}=\sqrt{\frac{8d}{d}} $ $ \Rightarrow $ $ \frac{{{T}_{1}}}{{{T}_{2}}}=2\sqrt{2} $ $ \Rightarrow $ $ {{T}_{2}}=\frac{{{T}_{1}}}{2\sqrt{2}}=\frac{24}{2\sqrt{2}} $ $ =6\sqrt{2}\,h $