Question

If the radius of the earth suddenly decreases by half of its present value, the time duration of one day will be (A) 6 hours

• A. 8 hours
• B. 12 hours
• C. 24 hours
• D. 48 hours
• E.

Hint:

If radius halves → time becomes one-fourth.

Answer 1

The Correct Option is A

Concept: Conservation of angular momentum
When no external torque acts on a rotating body: \[ I\omega = \text{constant} \]

Step 1: Relation between radius and moment of inertia
For a rotating body: \[ I \propto R^2 \] If radius is reduced to half: \[ R_2 = \frac{R_1}{2} \] \[ I_2 = I_1 \left(\frac{R_2}{R_1}\right)^2 = I_1 \left(\frac{1}{2}\right)^2 = \frac{I_1}{4} \]

Step 2: Apply conservation law
\[ I_1 \omega_1 = I_2 \omega_2 \] \[ I_1 \omega_1 = \frac{I_1}{4} \omega_2 \] \[ \Rightarrow \omega_2 = 4\omega_1 \]

Step 3: Relation between time period and angular velocity
\[ T = \frac{2\pi}{\omega} \Rightarrow T \propto \frac{1}{\omega} \]

Step 4: Find new time period
\[ T_2 = \frac{T_1}{4} \] \[ T_2 = \frac{24}{4} = 6 \text{ hours} \] Final Answer:
\[ \boxed{6 \text{ hours}} \] Conclusion:
Decrease in radius reduces moment of inertia, increasing angular speed and decreasing time period.