Question

If the earth were to suddenly contract to $\frac{1}{n}$th of its present radius without any change in its mass, the duration of the new day will be nearly:

• A. $\frac{24}{n}$ hours
• B. $24n$ hours
• C. $\frac{24}{n^2}$ hours
• D. $24n^2$ hours

Answer 1

Answer: (C)

$\frac{24}{n^2}$ hours

Using conservation of angular momentum
$I_1{\omega }_1=I_2{\omega }_2$

$\frac{2}{5}m{R}^2{\omega }_1=\frac{2}{5}m(\frac{R}{n}{)}^2{\omega }_2$

$\frac{{\omega }_1}{{\omega }_2}=\frac{1}{n^2}$

$\frac{{\tau }_1}{{\tau }_2}=n^2$

${\tau }_2=\frac{{\tau }_1}{n^2}$

${\tau }_1=24hrs$

${\tau }_2=\frac{24}{n^2}hrs$