Suppose the earth suddenly shrinks in size, still remaining spherical and mass unchanged (All gravitational forces pass through the centre of the earth).

Options

A. The days will become shorter.
B. The kinetic energy of rotation about its own axis will decrease
C. The duration of the year will increase
D. The magnitude of angular momentum about its axis will increase.

Explanation

To solve the problem of what happens when the Earth suddenly shrinks in size while keeping its mass unchanged, we can analyze the situation using the principles of angular momentum, moment of inertia, and rotational motion.

### Step-by-Step Solution:

1. **Understanding Angular Momentum Conservation**:
- Angular momentum (L) is given by the formula:
\[ L = I \cdot \omega \]
where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.
- Since the mass of the Earth remains unchanged and it is shrinking, the angular momentum will be conserved.

2. **Moment of Inertia**:
- The moment of inertia \( I \) for a solid sphere is given by:
\[ I = \frac{2}{5} m r^2 \]
where \( m \) is the mass and \( r \) is the radius.
- As the Earth shrinks, the radius \( r \) decreases, leading to a decrease in the moment of inertia \( I \).

3. **Effect on Angular Velocity**:
- Since angular momentum is conserved:
\[ I_1 \cdot \omega_1 = I_2 \cdot \omega_2 \]
where \( I_1 \) and \( \omega_1 \) are the initial moment of inertia and angular velocity, and \( I_2 \) and \( \omega_2 \) are the new moment of inertia and angular velocity after shrinking.
- If \( I_2 < I_1 \) (moment of inertia decreases), then \( \omega_2 \) must increase to keep the product \( I \cdot \omega \) constant.

4. **Time Period of Rotation**:
- The time period \( T \) of rotation is given by:
\[ T = \frac{2\pi}{\omega} \]
- Since \( \omega \) increases, \( T \) must decrease. This means that the Earth will complete its rotation in a shorter period of time.

5. **Kinetic Energy of Rotation**:
- The rotational kinetic energy \( K \) is given by:
\[ K = \frac{1}{2} I \omega^2 \]
- As \( \omega \) increases and \( I \) decreases, the overall kinetic energy will increase because the increase in \( \omega^2 \) will dominate.

6. **Conclusion**:
- The days on Earth will become shorter due to the increase in angular velocity.
- The kinetic energy of the Earth's rotation will increase.
- The duration of the year (time taken to orbit the sun) is not directly affected by the change in size but is related to the orbital mechanics, which remains unchanged in this scenario.

### Final Statements:
- **Correct Statement**: The days will become shorter.
- **Incorrect Statements**: Kinetic energy of rotation decreases, duration of the year increases, and the magnitude of angular momentum decreases.