A metal rod of Young's modulus Y and coefficient of thermal...

A metal rod of Young's modulus $Y$ and coefficient of thermal expansion $a$ is held at its two ends such that its length remains invariant. If its temperature is raised by $t^{\circ} \mathrm{C}$, then the linear stress developed in it is [2011] (A) $\frac{\alpha t}{Y}$ (B) $Y \alpha t$ (C) $\frac{Y}{\alpha t}$ (D) $\frac{1}{Y \alpha t}$

A metal rod of Young's modulus $Y$ and coefficient of thermal expansion $a$ is held at its two ends such that its length remains invariant. If its temperature is raised by $t^{\circ} \mathrm{C}$, then the linear stress developed in it is [2011]
(A) $\frac{\alpha t}{Y}$
(B) $Y \alpha t$
(C) $\frac{Y}{\alpha t}$
(D) $\frac{1}{Y \alpha t}$

Key Concepts

Thermal_Stress

Thermal stress is the stress induced in a material when its expansion or contraction due to temperature change is restrained. In the absence of constraints, a material would freely expand or contract; however, when movement is restricted, stress develops internally to oppose the temperature-induced strain.

Constrained_Expansion

When a material with a given coefficient of thermal expansion and Young's modulus is prevented from expanding or contracting due to fixed boundaries, the thermal strain (which is the product of the coefficient and temperature change) is counteracted by an opposing elastic strain. This constraint leads directly to the development of an internal stress proportional to the product of Young's modulus and the free thermal strain.

Youngs_Modulus

Young's modulus is a measure of a material's stiffness that defines the relationship between tensile stress and tensile strain in the elastic (linear) region of deformation. It quantifies how much a material will deform under a certain applied load and is fundamental in describing elastic behavior.

Coefficient_of_Thermal_Expansion

The coefficient of thermal expansion indicates how much a material's length changes per unit length per degree change in temperature. It is a critical physical property that describes the sensitivity of the material's dimensions to temperature variations.

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