G L Mittal and TARUN MITTAL Solutions for Chapter: Elasticity, Exercise 3: FOR DIFFERENT COMPETITIVE EXAMINATIONS

A wire fixed at the upper end stretched by the length $l$ by applying a force $F$. The work done in stretching is :

The potential energy of a long spring stretched by $2 cm$ is $U$. When stretched by $10 \mathrm{cm}$, the potential energy of the spring will be :

A uniform metal rod of $2 {\mathrm{mm}}^2$ cross-section is heated from $0°\mathrm{C}$ to $20°\mathrm{C}$. The coefficient of linear expansion of the rod is $12\mathrm{x}{10}^{-6} °{\mathrm{C}}^{-1}$. It's Young’s modulus of elasticity is ${10}^{11} \mathrm{N} {\mathrm{m}}^{-2}$. The energy stored per unit volume of the rod is :

A metallic bar is heated from $0°C$ to $100°C$. If this bar is so held that it can neither expand nor bend, then the force developed is :

An iron bar of length $1.0 \mathrm{m}$ and cross-section $1.0 {\mathrm{cm}}^{2 }$ is heated from $0°C$ to $100°C$ and is so held that it can neither expand nor bend. If $\alpha ={10}^{-5}°C^{-1}$ and $Y = {10}^{11} \mathrm{N} {\mathrm{m}}^{-2}$, then the force developed is:

Two rods of different materials having coefficients of linear expansion ${\alpha }_1, {\alpha }_2$ and Young’s moduli $Y_1, Y_2$ respectively are fixed between two rigid massive walls. The rods are heated such that they undergo the same increase in temperature. There is no bending of the rods. If ${\alpha }_1 : {\alpha }_2 = 2 : 3$ the thermal stresses developed in the two rods are equal, provided $Y_1 : Y_2$ is equal to :

A wooden wheel of radius $R$ is made of two semi-circular parts (see figure). The two parts are held together by a ring made of a metal strip of cross-sectional area $A$ and length $L$. $L$ is slightly less than $2nR$. To fit the ring on the wheel, it is heated so that its temperature rises by $∆T$ and it just steps over the wheel. As it cools down to the surrounding temperature, it presses the semi-circular parts together. If the coefficient of linear expansion of the metal is a and it's Young’s modulus is $Y$, the force that one part of the wheel applies on the other part is :

The coefficient of linear expansion and Young’s modulus of copper are respectively $1.5$times and $0.5$ times those of iron. The ratio of the forces developed in copper and iron bars of identical lengths and cross-sections and held fixed, when heated through the same range of temperature, is :