B M Sharma Solutions for Chapter: Elasticity, Exercise 6: Exercises

The wires $A$ and $B$ shown in the figure are made of the same material, and have radii $r_A$ and $r_B$ respectively. The block between them has a mass $m.$ When the force $F$ is $mg/3,$ one of the wires breaks:

A sphere of radius $0.1 \mathrm{m}$ and mass $8\pi \mathrm{kg}$ is attached to the lower end of a steel wire of length $5 \mathrm{m}$ and diameter ${10}^{-3} \mathrm{m}.$ The wire is suspended from $5.22 \mathrm{m}$ high ceiling of a room. When the sphere is made to swing as a simple pendulum, it just grazes the floor at its lowest point. Young's modulus of steel is $2\times {10}^{11}{\mathrm{Nm}}^{-2}.$ Find the velocity of the sphere at the lowest position in $\mathrm{m}/\mathrm{s}.$ (Given: $g=10 \mathrm{m}/{\mathrm{s}}^2$)

(Take $\pi =3.14$)

The breaking stress for a metal is $7.8\times {10}^9{\mathrm{Nm}}^{-2}.$ The density of the metal is $7800 \mathrm{kg}{ \mathrm{m}}^{-3}.$ If $g=10 \mathrm{N}{ \mathrm{kg}}^{-1},$ Find the maximum length of the wire made of this metal which may be suspended without breaking.

The average depth of Indian ocean is about $3000 \mathrm{m}.$ Find the fractional compression $\frac{\Delta V}{V}$ of water of the bottom of thc occan. Givcn, bulk modulus of watcr is $2.2\times {10}^9{\mathrm{Nm}}^{-2}$ and density of water $=1000 \mathrm{kg}{ \mathrm{m}}^{-3}.$

A solid sphere of radius $R$ made of a material of bulk modulus $B$ is surrounded by a liquid in a cylindrical container. $A$ massless piston of area $A$ (the area of container is also $A$ ) floats un the surface of the liquid. When a mass $M$ is placed on the piston to compress the liquid, find the fractional change in radius of the sphere. (Given: $\frac{Mg}{AB}=0.3$)

A wire having a length $L$ and cross-sectional area $A$ is suspended at one of its ends from a ceiling. Density and Young's modulus of material of the wire are $\rho$ and $Y,$ respectively. Find its strain energy due to its own weight in $\mu J.$ (Given: $\frac{{\rho }^2{g}^{2}A{L}^3}{Y}=12\times {10}^{-6} \mathrm{J}$)

Find the elastic potential energy per unit volume of water (in $\times {10}^3{\mathrm{Jm}}^{-3}$) at a depth of $1 \mathrm{km}.$ Given, compressibility of water $=5\times {10}^{10}\mathrm{SI}$ units and density of water $=1000 \mathrm{kg}{ \mathrm{m}}^{-3}.$

A ring of radius $r$ (made of wire of density $\rho$ ) placed on a smooth horizontal surface is rotated about a stationary vertical axis passing through its centre and perpendicular to the plane of the ring as shown in figure. Determine the angular velocity in $\mathrm{rad}/\mathrm{s}$ at which the ring breaks. The wire breaks at a tensile stress \sigma. (given $\sigma /\rho =9 \mathrm{N}-\mathrm{m}/\mathrm{kg}$ and $r=1 \mathrm{m}$)