Question

If the radius of the earth suddenly decreases by half of its present value. Then the time duration of one day will be

• A. 6 h
• B. 8 h
• C. 12 h
• D. 24 h
• E. 48 h

Answer 1

Answer: (A)

As we know in the absence of external torque, angular momentum is constant so,
$L_{\text{initial }}=L_{\text{final }}$
$m_1\left(r_1\times v_1\right)=m_2\left(r_2\times v_2\right)$
since, $v=r\omega$
$\Rightarrow m_1{\omega }_1{r}_1^2=m_2{\omega }_2{r}_2^2$
For earth mass remain constant (i.e., $m_1=m_2$ )
$\Rightarrow {\omega }_1{r}_1^2={\omega }_2{r}_2^2\left(\because \omega =\frac{2\pi }{T}\right)$
$\Rightarrow \frac{r_1^2}{T_1}=\frac{r_2^2}{T_2}$
Given, $T_1=24h,r_1=R_1$
and $r_2=\frac{R_1}{2}\Rightarrow \frac{T_2}{24}=\frac{R_1^2}{R_1^{2}4}$
$\Rightarrow T_2=6h$