Question

A light uniform rod of Young's modulus $Y$, cross sectional area $A$, coefficient of linear expansion $\alpha$ and length $l_0$ is rigidly connected to support at one end and the other end of the rod is connected to the spring as shown in the figure. The temperature of the rod is increased by $△\theta$ with supports remaining fixed. Initially, the spring is in natural length position. Spring force on the rod acts uniformly over the cross section during elongation of the rod. Find the net elongation of the rod. (Assume thermal strain to be small).

• A. $\frac{K{l}_0\alpha △\theta }{(K+YA/l_0)}$
• B. $\frac{YA\alpha △\theta }{(K+YA/l_0)}$
• C. $YA\alpha △\theta$
• D. None of the above

Answer 1

The correct option is A $\frac{K{l}_0\alpha △\theta }{(K+YA/l_0)}$

The elastic deformation in Spring rod system $=$ Total thermal deformation in Rod

Force through elastic deformation $=(k+\frac{YA}{l_o})△x$ of Spring and Rod system

Thermal deformation $=l_o\alpha △\theta$

Total spring force through thermal deformation $=k{l}_o\alpha △\theta$

So, equating the forces,

$(k+\frac{YA}{l_o})△x=k{l}_o\alpha △\theta △x=\frac{k{l}_o\alpha △\theta }{(k+\frac{YA}{l_o})}$

$△x$ is the net elongation of the rod.