Question

Maximum possible mass of a greased needle of length $l$ floating on water surface of tension $T$ is $CTl$. Find value of $C$. (Take $g =10\, m / s ^{2}$ )

Answer 1

Let the mass of the needle be $m$. As the liquid surface is distorted, the surface tension forces acing on both sides of the needle make an angle $\theta$, say, with vertical as shown in the figure.

Since the forces acting on the needle are $F , F$ and mg, resolving the forces vertically for its equilibrium,
$\sum F _{ y }= F \cos \theta+ F \cos \theta- mg =0$
$\Rightarrow m =\frac{2 F \cos \theta}{ g }$ where, $F = T l$
$\therefore m =\frac{2 T l \cos \theta}{ g }$
For $m$ to be maximum, $\cos \theta=1$
$\therefore m _{\max }=\frac{2 T l}{ g }=\frac{2 T l}{10}=0.2 Tl$
$\Rightarrow C =0.2$