Question

A metallic wire of density $d$ floats horizontally in water. The maximum radius of the wire so that the wire may not sink will be (surface tension of water = $T$)

• A. $\sqrt{\dfrac{2T}{\pi d g}}$
• B. $\sqrt{\dfrac{2\pi T}{d g}}$
• C. $\sqrt{\dfrac{2\pi T g}{d}}$
• D. $\sqrt{2\pi T g d}$

Hint:

A thin wire can float on water due to surface tension even if its density is greater than water—balance surface tension force with weight per unit length.

Answer 1

The Correct Option is A

Step 1: Consider a metallic wire of radius $r$ floating horizontally on water.
Step 2: The upward force due to surface tension acts along the two sides of the wire: \[ F_{\text{up}} = 2T \] (per unit length of the wire)
Step 3: The downward force is the weight of the wire per unit length: \[ F_{\text{down}} = \pi r^2 d g \]
Step 4: For the wire to be just on the verge of sinking, these forces must balance: \[ 2T = \pi r^2 d g \]
Step 5: Solve for $r$: \[ r^2 = \frac{2T}{\pi d g} \] \[ r = \sqrt{\frac{2T}{\pi d g}} \]
Step 6: Hence, the maximum radius of the wire so that it does not sink is: \[ \boxed{\sqrt{\dfrac{2T}{\pi d g}}} \]