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

Answer: (C)

6 $\sqrt{2}$

Answer 2

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, ${{d}_{1}}=d,\,{{d}_{2}}=8d$ $\Rightarrow$ $\frac{{{T}_{1}}}{{{T}_{2}}}=\sqrt{\frac{8d}{d}}$ $\Rightarrow$ $\frac{{{T}_{1}}}{{{T}_{2}}}=2\sqrt{2}$ $\Rightarrow$ ${{T}_{2}}=\frac{{{T}_{1}}}{2\sqrt{2}}=\frac{24}{2\sqrt{2}}$ $=6\sqrt{2}\,h$