The Value of g on the surface of earth is smallest at the equator because

Options

A. The centripetal force maximum at equator
B. The centripetal force is least at equator
C. The centrifugal force is maximum at equator
D. The angular speed of earth is least at equator

Answer

To solve the question regarding why the value of g on the surface of the Earth is smallest at the equator, we can break down the explanation into a series of logical steps.

### Step-by-Step Solution:

1. **Understanding the Concept of g**:
The acceleration due to gravity (g) at any point on the Earth's surface is influenced by two main factors: the mass of the Earth and the distance from the center of the Earth.

2. **Variation of g with Latitude**:
The formula for the effective acceleration due to gravity (g') at a latitude (λ) is given by:
\[ g' = g - R \omega^2 \cos^2(\lambda) \]
where:
- \( g \) is the standard acceleration due to gravity (approximately 9.81 m/s²),
- \( R \) is the radius of the Earth,
- \( \omega \) is the angular velocity of the Earth's rotation,
- \( \lambda \) is the latitude.

3. **Latitude at the Equator**:
At the equator, the latitude (λ) is 0 degrees. Therefore, the cosine of 0 degrees is 1:
\[ \cos^2(0) = 1 \]

4. **Centrifugal Force at the Equator**:
The centrifugal force due to the Earth's rotation is maximum at the equator because the radius (R) from the center of the Earth to the equator is the largest. This centrifugal force acts outward and reduces the effective weight of objects.

5. **Calculating g at the Equator**:
Substituting λ = 0 into the formula:
\[ g' = g - R \omega^2 \cdot 1 \]
This shows that the effective gravity (g') at the equator is reduced by the maximum centrifugal force.

6. **Comparison with Poles**:
At the poles (λ = 90 degrees), the centrifugal force is zero because the cosine of 90 degrees is zero:
\[ g' = g - R \omega^2 \cdot 0 = g \]
Thus, the value of g is maximum at the poles and minimum at the equator.

7. **Conclusion**:
Therefore, the value of g is smallest at the equator primarily due to the maximum centrifugal force acting on the objects at that location, which reduces the effective acceleration due to gravity.

### Final Answer:
The value of g on the surface of the Earth is smallest at the equator because the centrifugal force due to Earth's rotation is maximum at the equator, reducing the effective value of g.