Question

Mass remaining the same if the earth shrinks until its radius becomes half of its value, then the duration of new day would be

• A. 18 hrs
• B. 6 hrs
• C. 48 hrs
• D. 12 hrs

Answer 1

The correct option is B 6 hrs

Given: mass remaining the same if the earth shrinks until its radius becomes half of its value,

To find the duration of new day

Solution:

Since no external torque acts on the object, conservation of momentum can be applied to this problem.

Apply conservation of momentum, by considering earth as a solid sphere,

$I_1{\omega }_1=I_2{\omega }_2$

Since earth is a solid sphere

$I_1=\frac{2}{5}M{R}^2$

Hence

$\frac{I_1}{I_2}=\frac{\frac{2}{5}M{R}^2}{\frac{2}{5}M{\left(\frac{R}{2}\right)}^2}⟹\frac{I_1}{I_2}=4$

We know,

$\frac{{\omega }_2}{{\omega }_1}=\frac{T_1}{T_2}⟹\frac{I_1}{I_2}=\frac{{\omega }_2}{{\omega }_1}=4$

Substituting

$T_1=24$,

we have

$\frac{{\omega }_2}{{\omega }_1}=\frac{T_1}{T_2}⟹4=\frac{24}{T_2}⟹T_2=6hrs$

Hence the duration of new day is 6hours