Question

A compressive force, F is applied at the two ends of a long thin steel rod. It is heated, simultaneously, such that its temperature increases by ΔT. The net change in its length is zero. Let L be the length of the rod, A is its area of cross-section. Y is Young’s modulus, and α is its coefficient of linear expansion. Then, F is equal to

• (A) ${\mathrm{L}}^2\mathrm{Y}\mathrm{a}\mathrm{\Delta }\mathrm{T}$
• (B) $\frac{\mathrm{AY}}{\mathrm{a}\mathrm{\Delta }\mathrm{T}}$
• (C) $\mathrm{AYa}\mathrm{\Delta }\mathrm{T}$
• (D) $\mathrm{LAYa}\mathrm{\Delta }\mathrm{T}$

Answer 1

The Correct Option is C

Explanation:
Thermal expansion, $\mathrm{\Delta }\mathrm{L}=\mathrm{La}\mathrm{\Delta }\mathrm{T}$.....(i)Compression $\mathrm{\Delta }{\mathrm{L}}^{\prime }$ produced by applied force is given by,$\mathrm{Y}=\frac{\mathrm{F}\mathrm{L}}{\mathrm{A}\mathrm{\Delta }{\mathrm{L}}^{\prime }}\Rightarrow \mathrm{F}=\frac{\mathrm{Y}\mathrm{A}\mathrm{\Delta }{\mathrm{L}}^{\prime }}{\mathrm{L}}$......(ii)Net change in length $=0\Rightarrow \mathrm{\Delta }{\mathrm{L}}^{\prime }=\mathrm{\Delta }\mathrm{L}$......(iii)From equation (i),(ii) and (iii) we get,$\mathrm{F}=\mathrm{Y}\mathrm{A}\times (\mathrm{L}\mathrm{a}\mathrm{\Delta }\mathrm{T})\mathrm{L}=\mathrm{Y}\mathrm{A}\mathrm{a}\mathrm{\Delta }\mathrm{T}$Hence, the correct option is (C).