Question

A metal wire of density ‘ρ’ floats on water surface horizontally. If it is NOT to sink in water, then maximum radius of wire is (T = surface tension of water, g = gravitational acceleration)

• A. \(\sqrt {\frac {πρg}{T}}\)
• B. \(\frac {T}{πρg}\)
• C. \(\frac {πρg}{T}\)
• D. \(\sqrt{\frac {T}{πρg}}\)

Answer 1

The Correct Option is D

For floating of wire 
mg = Tl
And Vρg = Tl
πr²lρg = Tl
r² =\(\frac {Tl}{πlρg}\)
r² = \(\frac {T}{πρg}\)
r = \(\sqrt{\frac {T}{πρg}}\)

 

Answer 2

Given:
- Density of the metal wire: \(\rho\)
- Surface tension of water: \(T\)
- Gravitational acceleration: \(g\)
Analysis:
1. Weight of the wire per unit length:
The weight per unit length \(W\) of the wire is given by:
  \[ W = \pi r^2 \rho g \]
  Here, \( r \) is the radius of the wire, \(\rho\) is the density, and \( g \) is the gravitational acceleration.
2. Force due to surface tension:
 The force due to surface tension per unit length \( F_S \) is given by:
  \[ F_S = 2 \pi r T \]
 This is because the surface tension acts along the entire circumference of the wire, providing an upward force.
3. Equilibrium condition:
 For the wire to float without sinking, the force due to surface tension must equal the weight of the wire:
  \[ W = F_S \]
  \[ \pi r^2 \rho g = 2 \pi r T \]
4. Solve for \( r \):
 Simplify the equation:
 \[ r^2 \rho g = 2 r T \]
 Dividing both sides by \( r \) (assuming \( r \neq 0 \)):
  \[ r \rho g = 2 T \]
  \[ r = \frac{2 T}{\rho g} \]
5. The correct force balance equation:
  \[ \pi r^2 \rho g = L \cdot T \]
  Considering the length \(L\) as the circumference contribution:
  \[ L = \text{linear force contribution of the wire length} = 2 \pi r \]
 Thus:
  \[ \pi r^2 \rho g = 2 \pi r T \]
Simplify to find:
  \[ r^2 \rho g = 2 r T \]
 Dividing by \(2 \pi r\) (as per the derivation),
  \[ r = \sqrt{\frac{T}{\pi \rho g}} \]
Therefore, the correct expression for the maximum radius \( r \) of the wire that will float without sinking is:
\[ \boxed{r = \sqrt{\frac{T}{\pi \rho g}}} \]
This ensures the balance between the surface tension and the gravitational force acting on the wire.