A rod of length ℓ 0 having co-efficient of linear expansion α is joined as in figure. Initially all strings are in natural length and friction is absent. Temperature of rod is increased by Δ T . Col

Question

A rod of length ℓ 0 ​ having co-efficient of linear expansion α is joined as in figure. Initially all strings are in natural length and friction is absent. Temperature of rod is increased by Δ T . Column I Column II (A) Compression in spring I. (P) ℓ 0 ​ α Δ T (B) Compression in spring II. (Q) 3 2 ℓ 0 ​ ​ α Δ T (C) Compression in spring III. (R) 9 ℓ 0 ​ ​ α Δ T (D) Elongation in rod. (S) 9 2 ℓ 0 ​ ​ α Δ T (T) 10 3 ℓ 0 ​ ​ α Δ T

TestHub · Accepted Answer

Combine springs I and II in series; their equivalent stiffness is 2K/3. Spring III has stiffness K/3.
$Δ ℓ = ℓ_{0} α Δ T = x_{12} + x_{3}$
$\frac{2 K}{3} x_{12} = \frac{K}{3} x_{3} \Rightarrow x_{3} = 2 x_{12}$
$x_{12} = \frac{Δ ℓ}{3}, x_{3} = \frac{2Δ ℓ}{3}$
The two series springs have stiffness K and 2K.
$x_{I} + x_{II} = \frac{Δ ℓ}{3}, K x_{I} = 2 K x_{II}$
$x_{I} = \frac{2Δ ℓ}{9}, x_{II} = \frac{Δ ℓ}{9}$
Thus A-S, B-R, C-Q, D-P.

Column I	Column II
(A) Compression in spring I.	(P) $ℓ_{0} α Δ T$
(B) Compression in spring II.	(Q) $\frac{2 ℓ _{0}}{3} α Δ T$
(C) Compression in spring III.	(R) $\frac{ℓ _{0}}{9} α Δ T$
(D) Elongation in rod.	(S) $\frac{2 ℓ _{0}}{9} α Δ T$
	(T) $\frac{3 ℓ _{0}}{10} α Δ T$

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