Elongation Due to Self-Weight MCQ with answer and solution

Concept

Elongation Due to Self-Weight

Formula

ΔL = (γ × L²) / (2 × E)

Step-by-step solution

1. Use the formula for elongation due to self-weight: ΔL = (γ × L²) / (2 × E). 2. Observe that elongation (ΔL) is directly proportional to the square of the length (L²). 3. Substitute the new length L' = 2L into the formula: ΔL' = γ × (2L)² / (2 × E). 4. Simplify: ΔL' = 4 × \[(γ × L²) / (2 × E)\] = 4 × ΔL. 5. Conclude that the total elongation increases by 4 times.

Explanation

The elongation of a bar under its own weight is calculated using the formula: Elongation = (Weight Density × Length²) / (2 × Modulus of Elasticity) This formula shows that elongation depends on the square of the bar's length (Length²) and is independent of its diameter or cross-sectional area. When all dimensions (including length) are doubled: New Elongation ∝ (2 × Length)² = 4 × Length² Therefore, doubling the dimensions increases the total elongation by 4 times.

Shortcut

Elongation due to self-weight depends only on length squared (ΔL ∝ L²). Since dimensions double (n = 2), elongation increases by 2² = 4 times.

Common mistake

Students often mistakenly apply the formula for elongation due to an external axial force :Delta L = {PL}/{AE} or assume elongation is directly proportional to volume/weight, leading them to select 8 times or 16 times by failing to account for the quadrupling of the cross-sectional area.