$A$ wire of length $25 \ cm$ and radius $2 \ mm$ is fixed at one end. If a torque is applied at the other end to produce an angular displacement of $45^o$,calculate the work done in $J$. (Given: $\eta = 8 \times 10^{10} \ N/m^2$)

$A$ rubber band catapult has an initial length of $2 \, cm$ and a cross-sectional area of $5 \, mm^2$. It is stretched by an additional $2 \, cm$ and then released to project a stone of mass $20 \, g$. The velocity of the projected stone is (Young's modulus of rubber $= 5 \times 10^8 \, N/m^2$) (in $ \, m/s$)

$A$ wire is suspended by one end. At the other end,a weight equivalent to $20\, N$ force is applied. If the increase in length is $1.0\, mm$,the ratio of the increase in energy of the wire to the decrease in gravitational potential energy when the load moves downwards by $1\, mm$ will be:

The work done per unit volume to stretch a wire by $1\%$ of its length,having a cross-sectional area of $1\,mm^2$,is: $[Y = 9 \times 10^{11}\,N/m^2]$

Work done by the restoring force in a string within the elastic limit is $-10 \, J$. The maximum amount of heat produced in the string is .......... $J$.

$A$ metallic rod of length $L$ and cross-sectional area $A$ is made of a material of Young's modulus $Y$. If the rod is elongated by an amount $y$,then the work done is proportional to ......