Different minerals are being mined from within the earth and multi - storeyed complexes are being constructed. Due to this activity theoretically

Options

A. Angular speed of earth would increase
B. Angular momentum would increase
C. Time period of earth would decrease
D. Length of day would increase

Answer

**Step-by-Step Solution:**

1. **Understanding the Problem:**
The question states that minerals are being mined from the Earth, and multi-storeyed complexes are being constructed. We need to analyze the theoretical effects of these activities on the Earth's rotation.

2. **Concept of Mass Distribution:**
When minerals are mined, the mass that was originally located deep within the Earth is now being transferred to the surface. This alters the distribution of mass relative to the Earth's axis of rotation.

3. **Moment of Inertia (I):**
The moment of inertia (I) of a rotating body is given by the formula:
\[ I = \sum m r^2 \]
where \(m\) is the mass and \(r\) is the distance from the axis of rotation. By moving mass away from the axis of rotation (as we do when building structures), the moment of inertia increases.

4. **Conservation of Angular Momentum:**
Angular momentum (L) is conserved in the absence of external torques. The relationship can be expressed as:
\[ L = I \omega \]
where \( \omega \) is the angular velocity. Therefore, if the moment of inertia increases (due to mass being moved away from the axis), the angular velocity must decrease to keep the angular momentum constant.

5. **Effect on Angular Velocity:**
Since \(L\) is conserved, we can write:
\[ I_1 \omega_1 = I_2 \omega_2 \]
If \(I_2 > I_1\), then it follows that \(\omega_2 < \omega_1\). This means the angular velocity of the Earth decreases.

6. **Impact on Time Period:**
The time period \(T\) of rotation is related to angular velocity by the equation:
\[ T = \frac{2\pi}{\omega} \]
If \(\omega\) decreases, then \(T\) must increase. This means the length of the day will increase.

7. **Conclusion:**
Based on the analysis:
- The angular velocity of the Earth decreases (Option 1 is correct).
- The angular momentum remains constant (Option 2 is incorrect).
- The time period of the Earth increases (Option 3 is incorrect).
- The length of the day increases (Option 4 is correct).

**Final Answers:**
- Correct options are 1 and 4.

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