Question 207 of 223

If the earth suddenly shrinks to $\dfrac{1}{64}th$ of its original volume with its mass remaining the same, the period of rotation of earth becomes $\dfrac{24}{x}h$. The value of $x$ is ________

Your answerCheck answer

Show answer and solution▾

Answer: 16

$L=$ constant; $L=I\omega$
From the conservation of angular momentum,
$\dfrac{2}{5}M{R}^{2}{\omega }_{1}=\dfrac{2}{5}M{\left(\dfrac{R}{4}\right)}^{2}{\omega }_{2} \Rightarrow M{R}^{2}{\omega }_{1}=\dfrac{M{R}^{2}{\omega }_{2}}{16} \Rightarrow \dfrac{{\omega }_{1}}{{\omega }_{2}}=\dfrac{1}{16}$
Using $\omega =\dfrac{2\pi }{T}$ for each case along with the respective angular frequencies,
$\dfrac{{T}_{2}}{{T}_{1}}=\dfrac{1}{16}$
$\Rightarrow \dfrac{{T}_{1}}{{T}_{2}}=\dfrac{16}{1}$
Using ${T}_{1}=24h,$$x=16$ by comparing with the given time period in the question.