A solid hemisphere of radius a and weight W is floating in liquid and at a point on the base at a distance c from the centre rests a weight w. The tangent of the inclination of the axis of the hemisphere to the vertical for the corresponding position of equilibrium is . [assuming the base of the hemisphere to be entirely out of the fluid]

Options

A. `4/3c/aw/W`
B. `2/3c/aw/W`
C. `8/3c/a w/W`
D. `8/5 c/aw/W`

Answer

To solve the problem of finding the tangent of the inclination of the axis of the hemisphere to the vertical when a weight \( w \) is placed at a distance \( c \) from the center of the base of a solid hemisphere floating in a liquid, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the System**:
- We have a solid hemisphere of radius \( a \) and weight \( W \) floating in a liquid. The base of the hemisphere is entirely out of the fluid. A weight \( w \) is placed at a distance \( c \) from the center of the base.

2. **Identify the Center of Mass**:
- The center of mass of the solid hemisphere is located at a distance of \( \frac{3a}{8} \) from the flat base along the vertical axis.

3. **Set Up the Torque Equation**:
- For the system to be in equilibrium, the net torque about the point of contact (O) must be zero. The torques due to the weights \( W \) and \( w \) must balance each other.
- The torque due to the weight \( W \) (acting downwards at the center of mass) can be expressed as:
\[ \text{Torque due to } W = W \cdot \left(\frac{3a}{8} \sin \theta\right) \]
- The torque due to the weight \( w \) (acting at a distance \( c \) from the center) can be expressed as:
\[ \text{Torque due to } w = w \cdot (c \sin \theta) \]

4. **Set the Torques Equal**:
- Since the system is in equilibrium, we can set the two torques equal to each other:
\[ W \cdot \left(\frac{3a}{8} \sin \theta\right) = w \cdot (c \sin \theta) \]

5. **Simplify the Equation**:
- We can cancel \( \sin \theta \) from both sides (assuming \( \theta \neq 0 \)):
\[ W \cdot \frac{3a}{8} = w \cdot c \]

6. **Solve for \( \tan \theta \)**:
- Rearranging the equation gives:
\[ \tan \theta = \frac{8}{3} \cdot \frac{c \cdot w}{W \cdot a} \]

7. **Final Result**:
- Thus, the tangent of the inclination of the axis of the hemisphere to the vertical is given by:
\[ \tan \theta = \frac{8}{3} \cdot \frac{c \cdot w}{W \cdot a} \]