$A$ ring of a thin wire of cross-sectional area $a$ and length $l$ is dipped into a liquid of surface tension $\sigma$ and taken out so that a film of liquid is formed in the ring. If Young's modulus of the material of the wire is $Y$,then the longitudinal strain developed in the ring will be:

$A$ uniformly tapering conical wire is made from a material of Young's modulus $Y$ and has a normal,unextended length $L$. The radii at the upper and lower ends of this conical wire have values $R$ and $3R$,respectively. The upper end of the wire is fixed to a rigid support and a mass $M$ is suspended from its lower end. The equilibrium extended length of this wire would be:

If the ratio of diameters,lengths and Young's moduli of steel and brass wires shown in the figure are $p, q$ and $r$ respectively,then the corresponding ratio of increase in their lengths would be

When a wire of length $L$ clamped at one end is pulled by a force $F$ from the other end,its length increases by $l$. If the radius of the wire and the applied force were halved,then the increase in its length is

$A$ wooden wheel of radius $R$ is made of two semicircular parts. The two parts are held together by a metal ring of cross-sectional area $S$ and length $L$. $L$ is slightly less than $2\pi R$. To fit the ring on the wheel,the ring is heated by a temperature $\Delta T$. If the coefficient of linear expansion of the metal is $\alpha$ and its Young's modulus is $Y$,the force that one part of the wheel exerts on the other is:

Two separate wires $A$ and $B$ are stretched by $2 \, mm$ and $4 \, mm$ respectively,when they are subjected to a force of $2 \, N$. Assume that both the wires are made up of the same material and the radius of wire $B$ is $4$ times that of the radius of wire $A$. The lengths of the wires $A$ and $B$ are in the ratio of $a : b$. Then $a / b$ can be expressed as $1 / x$ where $x$ is: