A uniform rod of length L has a mass per unit length λand area of cross-section A . If the young

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A uniform rod of length L has a mass per unit length λand...

A uniform rod of length $L$ has a mass per unit length $\lambda$ and area of cross-section $A .$ If the young's modulus of the rod is $Y$, then the elongation in the rod due to its own weight is (a) $\frac{2 \lambda g L^{2}}{\Lambda Y}$ (b) $\frac{\lambda g L^{2}}{\Lambda Y}$ (c) $\frac{\lambda g L^{2}}{4 \Lambda Y}$ (d) $\frac{\lambda g L^{2}}{2 \Lambda Y}$

A uniform rod of length $L$ has a mass per unit length $\lambda$ and area of cross-section $A .$ If the young's modulus of the rod is $Y$, then the elongation in the rod due to its own weight is
(a) $\frac{2 \lambda g L^{2}}{\Lambda Y}$
(b) $\frac{\lambda g L^{2}}{\Lambda Y}$
(c) $\frac{\lambda g L^{2}}{4 \Lambda Y}$
(d) $\frac{\lambda g L^{2}}{2 \Lambda Y}$

Key Concepts

Young's Modulus

Young's Modulus is a material property that relates stress (force per unit area) to strain (relative deformation) in the linear elastic regime. It is used to determine how much a material will deform under a given load, and it plays a crucial role in calculating elongation and compression in structures subjected to forces.

Stress and Strain Relationship

The relationship between stress and strain, governed by Hooke's Law, provides a linear proportionality in elastic materials. When a force is applied to a material, the resulting stress leads to a proportional strain based on the material's Young's Modulus, which is key to determining deformation characteristics such as elongation.

Distributed Load

A distributed load refers to a force that is spread continuously along the length of a structure rather than applied at a single point. In the context of a self-weight loaded rod, the gravitational force creates a variable load along the rod's length, affecting the internal stress and resulting in a non-uniform elongation.

Differential Element Analysis

Differential element analysis involves breaking down a continuous object into infinitesimally small segments to analyze and integrate the effects of a variable load. For a rod under its own weight, analyzing a differential element allows one to relate the local tensile force to the extension of that element using the stress-strain relationship.

Integration of Differential Contributions

Integration is used to sum the contributions of all the infinitesimal extensions along the entire length of the rod. By integrating the local strains, which vary due to the changing load along the rod, one can calculate the total elongation caused by the continuous, distributed force of the rod's own weight.

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