If the angular speed of the earth is doubled, the value of acceleration due to gravity ( g ) at the north pole

Options

A. Doubles
B. Becomes half
C. Remains same
D. Becomes zero

Answer

To solve the problem regarding the effect of doubling the angular speed of the Earth on the acceleration due to gravity (g) at the North Pole, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula**: The acceleration due to gravity (g) at a latitude θ is given by the formula:
\[ g = G - R \omega^2 \cos^2(\theta) \]
where:
- \( G \) is the gravitational constant,
- \( R \) is the radius of the Earth,
- \( \omega \) is the angular speed of the Earth,
- \( \theta \) is the latitude.

2. **Identify the North Pole**: At the North Pole, the latitude \( \theta = 90^\circ \). Therefore, \( \cos^2(90^\circ) = 0 \). This means that the centrifugal force due to the Earth's rotation does not affect the value of g at the North Pole.

3. **Calculate g at the North Pole**: Since \( \cos^2(90^\circ) = 0 \), the formula simplifies to:
\[ g = G \]
Thus, the acceleration due to gravity at the North Pole is simply equal to the gravitational constant \( G \).

4. **Effect of Doubling Angular Speed**: If the angular speed \( \omega \) is doubled, it does not affect the value of g at the North Pole because the term involving \( \cos^2(\theta) \) remains zero. Therefore, even if \( \omega \) changes, \( g \) at the North Pole remains:
\[ g = G \]

5. **Conclusion**: The value of acceleration due to gravity (g) at the North Pole remains unchanged regardless of the increase in angular speed.

### Final Answer:
The value of acceleration due to gravity (g) at the North Pole remains the same even if the angular speed of the Earth is doubled.

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