A uniform wire of length $L$ and radius $r$ is twisted by an angle $\alpha$. If modulus of rigidity of the wire is $\eta$, then the elastic potential energy stored in wire, is .........
A$\frac{\pi \eta r^4 \alpha}{2 L^2}$
B$\frac{\pi \eta r^4 \alpha}{4 L^2}$
C$\frac{\pi \eta r^4 \alpha^2}{4 L}$
D$\frac{\pi \eta r^4 \alpha^2}{2 L}$
Medium
Download our app for free and get started
C$\frac{\pi \eta r^4 \alpha^2}{4 L}$
c (c)
$U=$ work done
We know
Work done $=\frac{\pi S r^4 \phi^2}{4 L}$ $\left\{\begin{array}{l}\text { Where, } \\ \phi=\text { Angle of twist }=\alpha \\ S=\text { Modulus of rigidity }=4\end{array}\right.$
Substituting values
$U=\frac{\pi \eta r^4 \alpha^2}{4 L}$
Download our app
and get started for free
Experience the future of education. Simply download our apps or reach out to us for more information. Let's shape the future of learning together!No signup needed.*
A rigid bar of mass $15\,kg$ is supported symmetrically by three wire each of $2 \,m$ long. These at each end are of copper and middle one is of steel. Young's modulus of elasticity for copper and steel are $110 \times 10^9 \,N / m ^2$ and $190 \times 10^9 \,N / m ^2$ respectively. If each wire is to have same tension, ratio of their diameters will be $ ............$
The specific heat at constant pressure and at constant volume for an ideal gas are ${C_p}$ and ${C_v}$ and its adiabatic and isothermal elasticities are ${E_\varphi }$ and ${E_\theta }$ respectively. The ratio of ${E_\varphi }$ to ${E_\theta }$ is
A stone of mass $20\, {g}$ is projected from a rubber catapult of length $0.1\, {m}$ and area of cross section $10^{-6} \,{m}^{2}$ stretched by an amount $0.04\, {m}$. The velocity of the projected stone is $....\,m\,/s.$ (Young's modulus of rubber $=0.5 \times 10^{9}\, {N} / {m}^{2}$ )
A uniform metal rod of $2\,\,mm^2$ cross section fixed between two walls is heated from $0\,^oC$ to $20\,^oC$ . The coefficient of linear expansion of rod is $12\,\,\times\,\,10^{-6}\,/^oC$ . Its Young's modulus of elasticity is $10^{11}\,\,N/m^2$ . The energy stored per unit volume of rod will be ....... $J/m^3$
A metal wire having Poisson's ratio $1 / 4$ and Young's modulus $8 \times 10^{10} \,N / m ^2$ is stretched by a force, which produces a lateral strain of $0.02 \%$ in it. The elastic potential energy stored per unit volume in wire is [in $\left.J / m ^3\right]$
If $\rho $ is the density of the material of a wire and $\sigma $ is breaking stress, the greatest length of the wire that can hang freely without breaking is