A light uniform rod of Young's modulus Y, cross sectional area A, coefficient of linear expansion α and length ℓ 0 is rigidly connected to support at one end and the other end of the rod is con

Question

A light uniform rod of Young's modulus Y, cross sectional area A, coefficient of linear expansion $α$ and length $ℓ_{0}$ is rigidly connected to support at one end and the other end of the rod is connected to the spring of constant K as shown in the figure. The temperature of the rod is increased by $Δ θ$ 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)

Source Q16 diagram

Options:

Answer:

B

Solution:

Let the rod elongation be $x$ . The spring compresses by the same amount.

Elastic compression subtracts from free thermal expansion.

$K x = \frac{Y A}{ℓ _{0}} (ℓ_{0} α Δ θ - x)$

$(K + \frac{Y A}{ℓ _{0}}) x = Y A α Δ θ$

$x = \frac{Y A α Δ θ}{K + \frac{Y A}{ℓ _{0}}}$

TestHub · Accepted Answer

Let the rod elongation be x . The spring compresses by the same amount. Elastic compression subtracts from free thermal expansion. K x = ℓ 0 ​ Y A ​ ( ℓ 0 ​ α Δ θ − x ) ( K + ℓ 0 ​ Y A ​ ) x = Y A α Δ θ x = K + ℓ 0 ​ Y A ​ Y A α Δ θ ​

Physics - Heat & Thermo Question with Solution | TestHub

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