In Column-I, different configurations of a uniform rod of mass m , length L , cross-sectional area A and Young’s modulus Y are shown. Match Column-I with Column-II. Column-I Column-II (A) A uniform ro

Question

In Column-I, different configurations of a uniform rod of mass $m$ , length $L$ , cross-sectional area $A$ and Young’s modulus $Y$ are shown. Match Column-I with Column-II.

Column-I	Column-II
(A) A uniform rod hangs vertically from a fixed support.	(P) Total elongation: $\frac{m gL}{2 A Y}$
(B) A constant force $F = m g$ is applied to one end of a rod on a smooth surface.	(Q) Total elongation: $\frac{m gL}{A Y}$
(C) Equal and opposite forces $F = m g$ act at the two ends.	(R) Uniform stress develops in the rod.
(D) The rod rotates about one end in gravity-free space at $ω = 3 g / L$ .	(S) Non-uniform stress develops in the rod.
	(T) Elastic potential energy in the rod: $\frac{m ^{2} g ^{2} L}{6 A Y}$

TestHub · Accepted Answer

For A and B, stress varies linearly along the rod.

$Δ L = \frac{m gL}{2 A Y}$

$U = \frac{m ^{2} g ^{2} L}{6 A Y}$

Hence, A and B match P, S and T.

For C, the tension and stress are uniform.

$Δ L = \frac{m gL}{A Y}$

Hence, C matches Q and R.

For D, centrifugal loading gives non-uniform stress.

$Δ L = \frac{m ω ^{2} L ^{2}}{3 A Y}$

$ω^{2} = \frac{3 g}{L} \Rightarrow Δ L = \frac{m gL}{A Y}$

Hence, D matches Q and S.

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