A rod of uniform cross sectional area A and length L has a weight W. It is suspended vertically from a fixed support. If Young's modulus for rod is Y, then elongation produced in the rod is A) (WL)/(YA) B) (WL)/(2YA) C) (WL)/(4YA) D) (3WL)/(4YA)?

Question

Answer

Option:-. B is correct ...Here,

CSA = A

Length = L

Weight = W

Let’s assume Y is the Young’s modulus of the rod.

Let us consider an element length dx at a distance x from the top.

Now tension acting on the element dx is due to the weight of the rod below it.

Weight of the rod below dx is F = W(L- x)/L

Now for the length dx the elongation dy becomes

F/A = Y dy/dx

=> dy = Fdx/(YA)

=> dy = W(L- x)dx/(YLA)

So total elongation due to length L becomes

Answer

Simple... use c.o.m then length( L/2)
use
Y=FL/∆L.A (here F=mg=W)

Answer

Problem Statement:
A rod of uniform cross-sectional area A and length L is suspended vertically from a fixed support. The weight of the rod is W. The Young's modulus of the rod is Y. We need to find the elongation produced in the rod.

Solution:

Step 1: Conceptual Understanding
To solve this problem, we need to understand the concept of Young's modulus and its relation to stress and strain. Young's modulus (Y) is a measure of the stiffness or elasticity of a material. It relates the stress (force per unit area) applied to a material to the strain (change in length per unit length) produced in the material.

Step 2: Applying the Formula
The elongation produced in the rod can be determined using the formula:

Elongation (ΔL) = (Force × Length) / (Area × Young's modulus)

In this case, the force acting on the rod is equal to its weight (W). The length of the rod is L, the cross-sectional area is A, and Young's modulus is Y. Substituting these values into the formula, we get:

ΔL = (W × L) / (A × Y)

Step 3: Simplifying the Equation
To simplify the equation, we can cancel out the length factor:

ΔL = (W × L) / (A × Y) = (W / A) × (L / Y)

Step 4: Final Answer
Therefore, the elongation produced in the rod is given by the equation:

ΔL = (W / A) × (L / Y)

Comparing this equation with the given options, we can see that the correct answer is (WL)/(AY), which corresponds to option A.

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