Question

If the earth were to suddenly contract to half its present size, without any change in its mass, the duration of the new day be

• A. $18hours$
• B. $30hours$
• C. $6hours$
• D. $112hours$

Answer 1

The correct option is C $6hours$

Given,

$r_2=\frac{1}{2}{r}_1$

$m_1=m_2=m\left(say\right)$

$T_1=24hr$

By the conservation of angular momentum,

$I_1{\omega }_1=I_2{\omega }_2$. . . . . . . . .(1)

Where, $\omega =\frac{2\pi }{T}$ angular velocity of the earth

$I=\frac{2}{5}m{r}^2$ moment of inertia of earth.

From equation (1),

$\frac{2}{5}m{r}_1^2.\frac{2\pi }{T_1}=\frac{2}{5}m{r}_2^2.\frac{2\pi }{T_2}$

$r_1^2.\frac{1}{T_1}=r_2^2.\frac{1}{T_2}$

$\frac{T_2}{T_1}=(\frac{r_2}{r_1}{)}^2=(\frac{r_1/2}{r_1}{)}^2=\frac{1}{4}$

$T_2=T_1.\frac{1}{4}$

$T_2=\frac{24}{4}hr=6hr$

The correct option is C.