There are two bodies A and B of same mass. A is placed near equator of earth and B is placed at a height `h` above the pole of earth. if both the bodies weighs equally. Find `h` in terms of radius `R` of earth, angular speed `omega` of earth and `g` acceleration due to gravity close to earth.

Options

A. `R omega^2/(2g)`
B. `R^2 omega^2/(2g)`
C. `gR/omega^2`
D. `g/omega^2`

Answer

To solve the problem, we need to find the height \( h \) above the pole where body B weighs the same as body A located at the equator. We will use the concepts of gravitational acceleration and the effects of Earth's rotation.

### Step-by-Step Solution:

1. **Understanding the Weights**:
- The weight of body A at the equator can be expressed as:
\[ W_A = m \cdot g_A \]
- The weight of body B at height \( h \) above the pole can be expressed as:
\[ W_B = m \cdot g_B \]
- Since both bodies weigh equally, we have:
\[ W_A = W_B \implies m \cdot g_A = m \cdot g_B \]
- We can cancel the mass \( m \) from both sides:
\[ g_A = g_B \]

2. **Calculating Gravitational Acceleration**:
- The gravitational acceleration at the equator (body A) is affected by the centrifugal force due to Earth's rotation:
\[ g_A = g - \omega^2 R \]
- The gravitational acceleration at height \( h \) above the pole (body B) is:
\[ g_B = g \left(1 - \frac{2h}{R}\right) \]

3. **Setting the Accelerations Equal**:
- From the previous step, we equate \( g_A \) and \( g_B \):
\[ g - \omega^2 R = g \left(1 - \frac{2h}{R}\right) \]

4. **Simplifying the Equation**:
- Distributing \( g \) on the right side:
\[ g - \omega^2 R = g - \frac{2gh}{R} \]
- Now, we can cancel \( g \) from both sides:
\[ -\omega^2 R = -\frac{2gh}{R} \]

5. **Rearranging to Solve for \( h \)**:
- Multiplying both sides by \( -1 \):
\[ \omega^2 R = \frac{2gh}{R} \]
- Now, multiply both sides by \( R \):
\[ \omega^2 R^2 = 2gh \]
- Finally, solving for \( h \):
\[ h = \frac{\omega^2 R^2}{2g} \]

### Final Result:
The height \( h \) above the pole where body B weighs the same as body A at the equator is given by:
\[ h = \frac{\omega^2 R^2}{2g} \]