Elongation of a vertical bar due to self-weight\nA prismatic bar of length l and diameter d is rigidly fixed at the top and hangs freely under its own weight. If w is the weight per unit volume and E is Young's modulus, what is the total axial elongation of the bar caused solely by its own weight?

Introduction / Context:
This problem tests the fundamentals of strength of materials related to axial deformation under a nonuniform load. A hanging bar experiences a linearly varying axial stress due to its own weight, which produces a definite elongation that must be accounted for in tall members, tie rods, and hangers.

Given Data / Assumptions:

• Prismatic bar: constant cross-sectional area A = (π/4) * d^2.
• Length = l; weight per unit volume = w (units N/m^3).
• Material follows Hooke’s law with Young’s modulus E; small deformations.
• Bar fixed at the top and hanging vertically in air; self-weight only.

Concept / Approach:
The axial force at a section is the weight of the portion below that section. Hence stress varies linearly along the length. Total elongation equals the integral of strain over the length, where strain = stress / E.

Step-by-Step Solution:
Let y be the distance measured downward from the top (y = 0 at top, y = l at bottom).Force at section y is the weight of the segment below: P(y) = w * A * (l − y).Stress at y: σ(y) = P(y) / A = w * (l − y).Strain at y: ε(y) = σ(y) / E = w * (l − y) / E.Elongation: δ = ∫[0→l] ε(y) dy = (w/E) * ∫[0→l] (l − y) dy = (w/E) * (l^2/2) = (w * l^2) / (2 * E).

Verification / Alternative check:
Dimensional check: w has N/m^3; multiplied by l^2 gives N/m; divide by E (N/m^2) gives meters, consistent with elongation.

Why Other Options Are Wrong:
Expressions without the 1/2 overestimate the elongation; terms involving d appear when using weight per unit length, not weight per unit volume; stating zero contradicts mechanics.

Common Pitfalls:
Confusing w as weight per unit length; forgetting that stress is not uniform along the bar; omitting the integral.

Final Answer:
(w * l^2) / (2 * E)