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 $g=10m/s^2$ )

Answer 1

Answer: 0.2

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,
$\sum F_y=F\cos \theta +F\cos \theta -mg=0$
$\Rightarrow m=\frac{2F\cos \theta }{g}$ where, $F=Tl$
$\therefore m=\frac{2Tl\cos \theta }{g}$
For $m$ to be maximum, $\cos \theta =1$
$\therefore m_{\max }=\frac{2Tl}{g}=\frac{2Tl}{10}=0.2Tl$
$\Rightarrow C=0.2$