A steel rod of 30 mm diameter and 0.3 m long is Subjected to a tensile force W kilo newton act

step 1 Hello students in this question we have a steel rod of diameter d equals to 30 m m and the lengt

A steel rod of 30 mm diameter and 0.3 m long is Subjected...

A steel rod of $30 \mathrm{~mm}$ diameter and $0.3 \mathrm{~m}$ long is Subjected to a tensile force $W$ kilo newton acting axially. The temperature of the rod raised through $80^{\circ} \mathrm{C}$ and total extension measured as $0.35 \mathrm{~mm}$. The value of $\mathrm{W}$ is: $\left(Y_{\mathrm{s}}=2 \times 10^{5} \mathrm{~N} / \mathrm{mm}^{2}, \alpha_{5}=12 \times 10^{-6} /{ }^{\circ} \mathrm{C}\right)$ (a) $29.20$ (b) 40 (c) 50 (d) 12

A steel rod of $30 \mathrm{~mm}$ diameter and $0.3 \mathrm{~m}$ long is Subjected to a tensile force $W$ kilo newton acting axially. The temperature of the rod raised through $80^{\circ} \mathrm{C}$ and total extension measured as $0.35 \mathrm{~mm}$. The value of $\mathrm{W}$ is:
$\left(Y_{\mathrm{s}}=2 \times 10^{5} \mathrm{~N} / \mathrm{mm}^{2}, \alpha_{5}=12 \times 10^{-6} /{ }^{\circ} \mathrm{C}\right)$
(a) $29.20$
(b) 40
(c) 50
(d) 12

Key Concepts

Axial Stress-Strain Relationship

The axial stress-strain relationship in a rod under tensile loading is derived from Hooke's law, where stress is proportional to strain with the proportionality constant being Young's modulus. This relationship is applied to calculate the force required to produce a given elastic deformation. When additional thermal expansion is involved, the measured total strain is a sum of the mechanical strain and the strain due to temperature changes, necessitating the separation of these effects to determine the true applied stress.

Mechanical Strain versus Thermal Strain

In many practical situations, a change in the dimensions of a material is due to both mechanical loading and thermal effects. Mechanical strain arises from applied forces and is governed by Hooke's law within the elastic limit. In contrast, thermal strain is produced by temperature changes and depends on the material's coefficient of thermal expansion. Differentiating between these two components is crucial for accurately analyzing net deformation when both forces and temperature variations are present.

Young's Modulus

Young's modulus is a measure of a material's stiffness, representing the relationship between tensile stress and tensile strain in the elastic region of the material's deformation. It provides a fundamental link between force and displacement in linearly elastic materials, allowing engineers to predict deformation under applied loads. In problems involving axial loading, knowing Young's modulus helps determine the amount of elastic elongation produced by an applied force.

Thermal Expansion

Thermal expansion is the phenomenon in which materials increase in length, area, or volume as temperature rises. In solids, this expansion is typically linear over moderate temperature ranges, and the coefficient of thermal expansion quantifies the relative change in length per degree change in temperature. This concept is significant when analyzing mechanical components that experience temperature fluctuations, as it affects dimensional tolerances and can add secondary strains.

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