Two rods of identical dimensions, with Young's modulus Y1...

Two rods of identical dimensions, with Young's modulus $Y_{1}$ and $Y_{2}$ are joined end to end. The equivalent Young's modulus for the composite rod is (A) $\frac{2 Y_{1} Y_{2}}{Y_{1}+Y_{2}}$ (B) $\frac{Y_{1} Y_{2}}{Y_{1}+Y_{2}}$ (C) $\frac{1}{2\left(Y_{1}+Y_{2}\right)}$ (D) $Y_{1}+Y_{2}$

Two rods of identical dimensions, with Young's modulus $Y_{1}$ and $Y_{2}$ are joined end to end. The equivalent Young's modulus for the composite rod is
(A) $\frac{2 Y_{1} Y_{2}}{Y_{1}+Y_{2}}$
(B) $\frac{Y_{1} Y_{2}}{Y_{1}+Y_{2}}$
(C) $\frac{1}{2\left(Y_{1}+Y_{2}\right)}$
(D) $Y_{1}+Y_{2}$

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Key Concepts

Young's Modulus

Young's modulus is a material property that measures the stiffness of a solid material by relating tensile stress to tensile strain. It quantifies how much a material will deform under a given load and is defined as the ratio of stress (force per unit area) over strain (relative deformation). This concept is fundamental in understanding elastic deformation in various mechanical and structural applications.

Stress-Strain Relationship

The stress-strain relationship describes how a material deforms under applied forces according to Hooke's law, up to the elastic limit. This relationship is key to determining how materials behave under load and is directly related to Young's modulus. It allows for the prediction of extension or compression in components when subjected to forces.

Series Combination of Elastic Elements

When elastic elements such as rods or springs are connected in series, their overall deformation is the sum of the individual deformations under the same applied force. The effective stiffness or modulus of the combined system is derived from the compliances (inverse of stiffness) of the individual elements. This concept is crucial in determining the equivalent properties of composite systems made of different materials or components.

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