## Two rods of different materials having coefficients of linear expansion α1 and α2 and Young’s moduli, Y1 and Y2, respectively…

Q: Two rods of different materials having coefficients of linear expansion α1 and α2 and Young’s moduli, Y1 and Y2 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 rods. If α12 = 2/3, then the thermal stresses developed in the two rods are equal provided Y1/Y2 is equal to

(a) 2 : 3

(b) 1 : 1

(c) 3 : 2

(d) 4 : 9

Ans: (c)

Sol: $\displaystyle Y_1 \alpha_1 \Delta \theta = Y_2 \alpha_2 \Delta \theta$ $\displaystyle Y_1 \alpha_1 = Y_2 \alpha_2$ $\displaystyle \frac{Y_1}{Y_2} = \frac{\alpha_2}{\alpha_1}$ $\displaystyle \frac{Y_1}{Y_2} = \frac{3}{2}$

## A thick rope of density ρ and length L is hung from a rigid support. The increase in length of the rope…

Q: A thick rope of density ρ and length L is hung from a rigid support. The increase in length of the rope due to its own weight is (Y is the Young’s modulus)

(a) $\displaystyle \frac{1}{4Y}\rho L^2 g$

(b) $\displaystyle \frac{1}{2Y}\rho L^2 g$

(c) $\displaystyle \frac{\rho L^2 g}{Y}$

(d) $\displaystyle \frac{\rho L g}{Y}$

Ans: (b)

Tension at mid is

T = Mg/2

Young’s modulus of Elasticity is $\displaystyle Y = \frac{T L}{A \Delta L}$ $\displaystyle \Delta L = \frac{T L}{A Y}$ $\displaystyle \Delta L = \frac{M g L}{2A Y}$ $\displaystyle \Delta L = \frac{(A L \rho ) g L}{2 A Y}$ $\displaystyle \Delta L = \frac{\rho g L^2}{2 Y}$

## A uniform pressure ‘P’ is exerted on all sides of a solid cube at temperature 0°C. In order to bring…

Q: A uniform pressure ‘P’ is exerted on all sides of a solid cube at temperature 0°C. In order to bring the volume of the cube to the original volume, the temperature of the cube must be increased by t°C. If α is the linear coefficient and K be the bulk modulus of the material of the cube, then t is equal to

(a) $\displaystyle \frac{3P}{K \alpha}$

(b) $\displaystyle \frac{P}{2 K \alpha}$

(c) $\displaystyle \frac{P}{3 K \alpha}$

(d) $\displaystyle \frac{P}{K \alpha}$

Ans: (c)

## When a rubber ball of volume v, bulk modulus K is at a depth h in water then decrease in its volume..

Q: When a rubber ball of volume V, bulk modulus K is at a depth h in water then decrease in its volume is

(a) $\displaystyle \frac{h \rho g V}{K}$

(b) $\displaystyle \frac{h \rho g V}{2K}$

(c) $\displaystyle \frac{h \rho g V}{K}$

(d) $\displaystyle \frac{h \rho g V}{4K}$

Ans: (a)
Bulk Modulus of Elasticity is $\displaystyle K = \frac{P}{-\Delta V/V}$ $\displaystyle -\Delta V = \frac{P V}{K}$ $\displaystyle = \frac{\rho g h V}{K}$

## A material has normal density ρ and bulk modulus K. The increase in the density of the material when it is…

Q: A material has normal density ρ and bulk modulus K. The increase in the density of the material when it is subjected to an external pressure P from all sides is

(a) $\displaystyle \frac{P}{\rho K}$

(b) $\displaystyle \frac{K P}{\rho }$

(c) $\displaystyle \frac{P \rho}{ K}$

(d) $\displaystyle \frac{K \rho}{P}$

Ans: (c) $\displaystyle \rho = \frac{M}{V}$ $\displaystyle \frac{\Delta \rho}{\rho} = \frac{-\Delta V}{V}$

Bulk Modulus of Elasticity is $\displaystyle K = \frac{P}{-\Delta V/V}$ $\displaystyle \frac{-\Delta V}{V} = \frac{P}{K}$ $\displaystyle \frac{\Delta \rho}{\rho} = \frac{P}{K}$ $\displaystyle \Delta \rho = \frac{P \rho}{K}$