A ring of radius r, and weight W is lying on a liquid surface . If the surface tension of the liquid is T , then the minimum force required to be applied in order to lift the ring up

Options

A. W
B. 2W
C. `W+4 pi rT`
D. `W+2 pi rT`

Explanation

To solve the problem of finding the minimum force required to lift a ring of radius \( r \) and weight \( W \) lying on a liquid surface with surface tension \( T \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Forces Acting on the Ring:**
The ring is subjected to two main forces:
- The weight of the ring \( W \) acting downwards.
- The upward force due to surface tension acting on the ring.

2. **Calculate the Length of the Ring in Contact with the Liquid:**
The total length of the ring in contact with the liquid surface is equal to its circumference. The circumference \( C \) of the ring is given by:
\[ C = 2\pi r \]
Since the ring is circular and the entire circumference is in contact with the liquid, the effective length in contact is:
\[ L = 2\pi r \]

3. **Determine the Force Due to Surface Tension:**
The force due to surface tension \( F_T \) acting on the ring can be calculated using the formula:
\[ F_T = T \times L \]
Substituting the length \( L \):
\[ F_T = T \times (2\pi r) = 2\pi r T \]

4. **Calculate the Total Downward Force:**
The total downward force acting on the ring is the sum of its weight \( W \) and the force due to surface tension \( F_T \):
\[ F_{\text{down}} = W + F_T = W + 2\pi r T \]

5. **Determine the Minimum Force Required to Lift the Ring:**
To lift the ring, the applied force \( F_{\text{applied}} \) must be equal to the total downward force:
\[ F_{\text{applied}} = W + 2\pi r T \]

### Final Result:
Thus, the minimum force required to lift the ring up is:
\[ F_{\text{applied}} = W + 2\pi r T \]