A metal wire of density `rho` floats on water surface horozontally. If is NOT to sink in water, then maximum radius of wire is proportional to (where, T=surface tension of water, g=gravitational acceleration)

Options

A. `sqrt((2T)/(pirhog))`
B. `sqrt((4T)/(pirhog))`
C. `sqrt((T)/(pirhog))`
D. `sqrt((Trho)/(pig))`

Answer

To solve the problem, we need to establish the relationship between the maximum radius of a metal wire that can float on the surface of water without sinking, given the surface tension of water (T), the density of the wire (ρ), and the gravitational acceleration (g).

### Step-by-Step Solution:

1. **Understanding the Forces**:
- When a metal wire floats on the water surface, it experiences an upward force due to surface tension and a downward force due to its weight.
- The upward force due to surface tension (F_t) can be expressed as:
\[ F_t = T \times L \]
where \(L\) is the length of the wire in contact with the water surface.

2. **Weight of the Wire**:
- The weight (W) of the wire can be expressed as:
\[ W = m \times g = \rho \times V \times g \]
where \(V\) is the volume of the wire, and \(\rho\) is the density of the wire.

3. **Volume of the Wire**:
- The volume of the wire can be expressed in terms of its cross-sectional area (A) and length (L):
\[ V = A \times L \]
- For a circular wire, the cross-sectional area \(A\) can be expressed as:
\[ A = \pi r^2 \]
where \(r\) is the radius of the wire.

4. **Setting Up the Equation**:
- At the equilibrium condition (when the wire is floating), the upward force due to surface tension equals the weight of the wire:
\[ T \times L = \rho \times (\pi r^2 \times L) \times g \]
- We can cancel \(L\) from both sides (assuming \(L \neq 0\)):
\[ T = \rho \pi r^2 g \]

5. **Solving for the Radius**:
- Rearranging the equation to solve for \(r^2\):
\[ r^2 = \frac{T}{\rho \pi g} \]
- Taking the square root to find \(r\):
\[ r = \sqrt{\frac{T}{\rho \pi g}} \]

6. **Proportionality**:
- From the final expression, we can conclude that the maximum radius \(r\) is proportional to:
\[ r \propto \sqrt{T} \]
- and inversely proportional to:
\[ r \propto \frac{1}{\sqrt{\rho g}} \]

### Conclusion:
The maximum radius of the wire that can float on water without sinking is given by:
\[ r \propto \sqrt{\frac{T}{\rho g}} \]