A uniform rod of length L has a mass per unit length λand area of cross-section A . The elongati

step 1 Hi friends here it is given L is the length of the rod. Lambda is mass per unit length. So mass

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 $A .$ The elongation in the rod is $l$ due to its own weight if it is suspended from the ceiling of a room. The Young's modulus of the rod is (A) $\frac{2 \lambda g L^{2}}{A l}$ (B) $\frac{\lambda g L^{2}}{2 A l}$ (C) $\frac{2 \lambda g L}{A l}$ (D) $\frac{\lambda g l^{2}}{A L}$

A uniform rod of length $L$ has a mass per unit length $\lambda$ and area of cross-section $A .$ The elongation in the rod is $l$ due to its own weight if it is suspended from the ceiling of a room. The Young's modulus of the rod is
(A) $\frac{2 \lambda g L^{2}}{A l}$
(B) $\frac{\lambda g L^{2}}{2 A l}$
(C) $\frac{2 \lambda g L}{A l}$
(D) $\frac{\lambda g l^{2}}{A L}$

Key Concepts

Mass per Unit Length

The mass per unit length represents how mass is distributed along the rod and is key to determining the gravitational force acting on any segment of the rod. This parameter is essential for developing the relationship between the applied gravitational forces and the resulting elongation, as it directly affects the calculation of the distributed load.

Cross-Sectional Area

The cross-sectional area of a rod is a critical factor in determining its stress under a given load. As stress is the force distributed over the area, the cross-sectional area appears in the equations relating force, deformation, and material properties, thereby influencing the overall elongation and the calculated Young's modulus.

Young's Modulus

Young's modulus is a fundamental property of materials that describes their stiffness, defined as the ratio of tensile stress (force per unit area) to tensile strain (relative deformation) in the elastic region. It quantitatively indicates how much a material will deform under a given load and is crucial in predicting the behavior of structural elements under stress.

Distributed Load Integration

When a rod is suspended by its own weight, the load is not applied at a single point but is distributed continuously along its length. This requires setting up an integral to sum the differential contributions of strain from each infinitesimal segment, taking into account that the force acting on each segment varies with its position along the rod.

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