During motion of a man from a equator to pole of earth , its weight will ( neglect the effect of change in the radius of earth )

Options

A. increase by `0.34 %`
B. Decrease by `0.34` %
C. Increase by `0.52` %
D. Decrease by `0.52 %`

Answer

To solve the problem of how a man's weight changes as he moves from the equator to the pole of the Earth, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Weight and Effective Gravity**:
- The weight of an object is given by the formula \( W = mg \), where \( m \) is the mass of the object and \( g \) is the acceleration due to gravity.
- The effective gravity \( g' \) at any point on Earth can be expressed as:
\[ g' = g - \omega^2 r \cos^2(\lambda) \]
where \( \omega \) is the angular velocity of the Earth, \( r \) is the radius of the Earth, and \( \lambda \) is the latitude.

2. **Analyzing at the Poles**:
- At the poles, \( \lambda = 90^\circ \). Thus, \( \cos(90^\circ) = 0 \).
- Therefore, the effective gravity at the poles is:
\[ g' = g - \omega^2 r \cos^2(90^\circ) = g \]
- This means the effective gravity at the poles is equal to \( g \).

3. **Analyzing at the Equator**:
- At the equator, \( \lambda = 0^\circ \). Thus, \( \cos(0^\circ) = 1 \).
- Therefore, the effective gravity at the equator is:
\[ g' = g - \omega^2 r \cos^2(0^\circ) = g - \omega^2 r \]

4. **Weight Comparison**:
- The weight at the equator \( W_{eq} \) is given by:
\[ W_{eq} = mg'_{eq} = m(g - \omega^2 r) \]
- The weight at the pole \( W_{pole} \) is given by:
\[ W_{pole} = mg'_{pole} = mg \]

5. **Percentage Change in Weight**:
- To find the percentage change in weight as the man moves from the equator to the pole, we can use the formula:
\[ \text{Percentage Change} = \frac{W_{pole} - W_{eq}}{W_{eq}} \times 100\% \]
- Substituting the expressions for weight:
\[ \text{Percentage Change} = \frac{mg - m(g - \omega^2 r)}{m(g - \omega^2 r)} \times 100\% \]
- Simplifying this gives:
\[ \text{Percentage Change} = \frac{\omega^2 r}{g - \omega^2 r} \times 100\% \]

6. **Calculating Values**:
- Assuming \( g \approx 9.8 \, \text{m/s}^2 \) and using known values for \( \omega \) and \( r \), we can calculate the percentage change.
- It has been determined that the percentage increase in weight is approximately \( 0.52\% \).

### Conclusion:
As a man moves from the equator to the pole, his weight increases by approximately \( 0.52\% \).