For a cantilever beam of span L, what is the ratio of free-end deflections due to the same isolated load P placed at L/3 and at 2L/3 from the fixed end?

Introduction / Context:
Deflection of beams under point loads depends on where the load acts. For a cantilever, the same load located nearer to the free end generally causes a larger free-end deflection than the same load applied closer to the fixed end. This question compares free-end deflections for a single cantilever under identical load magnitudes but at two different positions along the span.

Given Data / Assumptions:

• Prismatic cantilever beam with span L, flexural rigidity E*I constant.
• Point load P applied at a distance a from the fixed end.
• We compare deflection at the free end for a = L/3 and a = 2L/3.

Concept / Approach:
For a cantilever with a point load P applied at distance a from the fixed end, the free-end deflection is given by:
δ_free = P * a^2 * (3L - a) / (6 * E * I)This arises from integrating the curvature equation M / (E * I) along the span considering the load location. Since P, E, I, and L are the same for both cases, the ratio depends only on the factor a^2 * (3L - a).

Step-by-Step Solution:
Case 1 (a = L/3): factor = (L/3)^2 * (3L - L/3) = (L^2/9) * (8L/3) = 8L^3/27.Case 2 (a = 2L/3): factor = (2L/3)^2 * (3L - 2L/3) = (4L^2/9) * (7L/3) = 28L^3/27.Ratio δ(L/3) : δ(2L/3) = (8/27) : (28/27) = 8 : 28 = 2 : 7.

Verification / Alternative check:
Sanity check: load nearer the free end (2L/3) should cause greater free-end deflection; ratio < 1 confirms this expectation.

Why Other Options Are Wrong:

• 7 : 2 and 4 : 1: Invert the true ratio; contradict physical expectation.
• 1 : 4 and 3 : 8: Do not match the derived factor relationship.

Common Pitfalls:
Using end-load formula δ = P*L^3/(3*E*I) irrespective of load position; forgetting that only the segment from the load to the free end directly influences δ_free in this relation.

Final Answer:
2 : 7