If earth radius reduces to half of its present value then the time period of rotation will become:

Options

A. one fourth
B. one half
C. double
D. invariant

Answer

To solve the problem of how the time period of Earth's rotation changes when its radius is reduced to half, we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Conservation of Angular Momentum**:
- Since no external torque is acting on Earth, its angular momentum must be conserved. This means that the initial angular momentum (L) is equal to the final angular momentum.
- Mathematically, this can be expressed as:
\[ L = I \omega = \text{constant} \]
where \(I\) is the moment of inertia and \(\omega\) is the angular velocity.

2. **Calculate the Moment of Inertia**:
- For a solid sphere like Earth, the moment of inertia \(I\) is given by:
\[ I = \frac{2}{5} m r^2 \]
- If the radius \(r\) is reduced to half, the new radius \(r' = \frac{r}{2}\). The new moment of inertia \(I'\) will be:
\[ I' = \frac{2}{5} m \left(\frac{r}{2}\right)^2 = \frac{2}{5} m \frac{r^2}{4} = \frac{1}{10} m r^2 \]

3. **Set Up the Conservation of Angular Momentum Equation**:
- Before the change:
\[ L = I \omega = \frac{2}{5} m r^2 \omega \]
- After the change:
\[ L' = I' \omega' = \frac{1}{10} m r^2 \omega' \]
- Setting these equal gives:
\[ \frac{2}{5} m r^2 \omega = \frac{1}{10} m r^2 \omega' \]

4. **Simplify the Equation**:
- Cancel \(m\) and \(r^2\) (assuming \(r \neq 0\)):
\[ \frac{2}{5} \omega = \frac{1}{10} \omega' \]
- Rearranging gives:
\[ \omega' = 4 \omega \]

5. **Relate Angular Velocity to Time Period**:
- The angular velocity \(\omega\) is related to the time period \(T\) by:
\[ \omega = \frac{2\pi}{T} \]
- Thus, the new angular velocity \(\omega'\) can be expressed as:
\[ \omega' = \frac{2\pi}{T'} \]
- Substituting gives:
\[ 4 \frac{2\pi}{T} = \frac{2\pi}{T'} \]

6. **Solve for the New Time Period**:
- Cancel \(2\pi\) from both sides:
\[ 4 \frac{1}{T} = \frac{1}{T'} \]
- Rearranging gives:
\[ T' = \frac{T}{4} \]

7. **Conclusion**:
- Therefore, if the Earth's radius is reduced to half of its present value, the time period of rotation will become one-fourth of its original value.

### Final Answer:
The time period of rotation will become \( \frac{T}{4} \).

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