Question

Four rods A,B,C,D of same length and material but of different radii r,r√2,r√3 and 2r respectively are held between two rigid walls. The temperature of all rods is increased by same amount. If the rods do not bend, then

(A) The stress in the rods are in the ratio of 1:2:3:4.

(B) The force on the rod exerted by the wall are in the ratio of 1:2:3:4.

(C) The energy stored in the rods due to elasticity are in the ratio 1:2:3:4.

(D) The strains produced in the rods are in the ratio of 1:2:3:4.

(A) The stress in the rods are in the ratio of 1:2:3:4.

(B) The force on the rod exerted by the wall are in the ratio of 1:2:3:4.

(C) The energy stored in the rods due to elasticity are in the ratio 1:2:3:4.

(D) The strains produced in the rods are in the ratio of 1:2:3:4.

- all A,B,C and D are correct
- only A and C are correct
- only B and C are correct
- only A and D are correct

Solution

The correct option is **C** only B and C are correct

Thermal stress is, TA=Yα△T...(i)

Force exerted is, T=YAα△T⇒T∝A..(ii)

Energy stored is, E=12×(stress)2×AlY...(iii)

Strain is, ϵ=stressY...(iv)

From (i) and (iv), it is clear that stress and strain depend only on the material and △T which are same for all the rods. Thus their ratios are 1:1:1:1.

From (ii) and (iii) force and energy stored are proportional to area of cross-section A=πr2. Thus, their ratios are

πr2:π(√2r)2:π(√3r)2:π(2r)2=1:2:3:4.

Thermal stress is, TA=Yα△T...(i)

Force exerted is, T=YAα△T⇒T∝A..(ii)

Energy stored is, E=12×(stress)2×AlY...(iii)

Strain is, ϵ=stressY...(iv)

From (i) and (iv), it is clear that stress and strain depend only on the material and △T which are same for all the rods. Thus their ratios are 1:1:1:1.

From (ii) and (iii) force and energy stored are proportional to area of cross-section A=πr2. Thus, their ratios are

πr2:π(√2r)2:π(√3r)2:π(2r)2=1:2:3:4.

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