A thin metal disc of radius `r` floats on water surface and bends the surface downwards along the perimeter making an angle `theta` with vertical edge of the disc . If the disc displaces a weight of water `W` and surface tension of water is `T`, then the weight of metal disc is

Options

A. `2pirT=W`
B. `2pirT costheta-W`
C. `2pirT costheta+W`
D. `W-2pirT costheta`

Explanation

To solve the problem, we need to analyze the forces acting on the thin metal disc floating on the water surface. The disc bends the surface of the water and makes an angle θ with the vertical. We will use the principles of buoyancy and surface tension to find the weight of the metal disc.

### Step-by-Step Solution:

1. **Identify Forces Acting on the Disc**:
- The forces acting on the disc include:
- The buoyant force (FB) acting upwards.
- The surface tension force (T) acting along the perimeter of the disc, which has both vertical and horizontal components.

2. **Calculate the Surface Tension Force**:
- The surface tension force acts along the circumference of the disc. The total length of the perimeter (L) of the disc is given by:
\[ L = 2\pi r \]
- The vertical component of the surface tension force (acting upwards) can be calculated as:
\[ F_T = 2T \cos(\theta) \]
- Here, \(2T\) is the total surface tension acting along the perimeter, and \(\cos(\theta)\) gives the vertical component.

3. **Apply Archimedes' Principle**:
- According to Archimedes' principle, the buoyant force (FB) is equal to the weight of the water displaced by the disc. This is given as:
\[ F_B = W \]
- Where W is the weight of the water displaced.

4. **Set Up the Equilibrium Condition**:
- At equilibrium, the total upward forces must balance the weight of the disc (mg). Therefore, we can write:
\[ mg = F_B + F_T \]
- Substituting the expressions for \(F_B\) and \(F_T\):
\[ mg = W + 2T \cos(\theta) \]

5. **Rearranging to Find the Weight of the Disc**:
- Rearranging the equation gives us the weight of the metal disc:
\[ mg = W + 2T \cos(\theta) \]
- Thus, the weight of the metal disc is:
\[ mg = W + 2T \cos(\theta) \]

### Final Result:
The weight of the metal disc is given by:
\[ \text{Weight of the disc} = W + 2T \cos(\theta) \]