Question

A metal wire of length 'L' and density 'd' floats horizontally on the free surface of water. The maximum radius of the wire does not sink in water is
[ \(T\) = surface tension of water , \(g\) = gravitational acceleration]

• A. \(\sqrt{\frac{2dg}{πT}}\)
• B. \(\sqrt{\frac{2T}{πdg}}\)
• C. \(\sqrt{\frac{πdg}{2T}}\)
• D. \(\sqrt{\frac{T}{πdg}}\)

Hint:

The wire floats when the upward surface tension force equals its weight; the force acts along both sides of the wire.

Answer 1

The Correct Option is B

Step 1: Force per unit length
The total surface tension force on the wire is $T$ per unit length on each side, so for length $L$ it is $2TL$.

Step 2: Limit of floating
At the limit the wire is about to sink, so $\text{weight} = 2TL$.

Step 3: Write mass
Mass $= d\,\pi r^2L$, so weight $= d\,\pi r^2Lg$. Equate to $2TL$ and cancel $L$: $d\pi r^2g = 2T$.

Step 4: Result
$r = \sqrt{\dfrac{2T}{\pi d g}}$. A thicker wire has a bigger weight than the surface tension can support, so it sinks.

Final Answer:
The maximum radius is sqrt(2T/(pi d g)). This is option (B). \[ \boxed{\text{(B) }\sqrt{\frac{2T}{\pi d g}}} \]