Question

A steel wire of diameter $0.5\, mm$ and Young’s modulus $2\times10^{11} \,N\,m^{-2}$ carries a load of mass $M$. The length of the wire with the load is $1.0 \,m$. A vernier scale with $10$ divisions is attached to the end of this wire. Next to the steel wire is a reference wire to which a main scale, of least count $1.0\, mm$, is attached. The $10$ divisions of the vernier scale correspond to $9$ divisions of the main scale. Initially, the zero of vernier scale coincides with the zero of main scale. If the load on the steel wire is increased by $1.2\, kg$, the vernier scale division which coincides with a main scale division is __________. Take $g\, =\, 10\, ms^{-2}$ and $\pi$ =

Answer 1

$d=0.5\, mm $
$ Y=2 \times 10^{11}\, \,l=1\, m$
$\Delta l=\frac{F l}{A y}=\frac{\frac{m g l}{\pi d^{2}}}{4} y$
$=\frac{1.2 \times 10 \times 1}{\frac{\pi}{4} \times\left(5 \times 10^{-4}\right)^{2} \times 2 \times 10^{11}}$
$\Delta l=\frac{1.2 \times 10}{\frac{3.2}{4} \times 25 \times 10^{-8} \times 2 \times 10^{11}}$
$= \frac{12}{0.8 \times 25 \times 2 \times 10^{3}}=\frac{12}{40 \times 10^{3}}=0.3\, mm$
So $3^{rd}$ division of a vernier scale will coincide with main scale.