A simply supported beam (length L) has bending moment M(x) = a * (x - x^3 / L^2), 0 ≤ x < L, where a is a constant. At what location is the shear force zero?

Introduction / Context:
Shear force is the derivative of bending moment with respect to the beam axis. Locating where shear is zero helps identify positions of maximum or minimum bending moment and is fundamental in beam analysis.

Given Data / Assumptions:

• M(x) = a * (x - x^3 / L^2) for 0 ≤ x < L.
• a is a constant, L is span.
• Beam is simply supported (moment at supports is zero).

Concept / Approach:
Shear force V(x) is given by the first derivative of the bending moment: V(x) = dM/dx. Set V(x) = 0 and solve for x within the span to find stationary points of M(x).

Step-by-Step Solution:
M(x) = a(x - x^3/L^2).V(x) = dM/dx = a(1 - 3x^2/L^2).Set V(x) = 0 → 1 - 3x^2/L^2 = 0.3x^2/L^2 = 1 → x^2 = L^2/3 → x = L/√3 (taking the positive root in 0 < x < L).

Verification / Alternative check:
Second derivative test: dV/dx = -6a x / L^2. At x = L/√3, sign of dV/dx determines whether M has max/min, consistent with beam behavior.

Why Other Options Are Wrong:

• x = L/3 or L/2: do not satisfy V(x) = 0 for the given M(x).
• At supports: V at supports is generally not zero for this M(x); moreover M(x) is specified for 0 ≤ x < L.

Common Pitfalls:

• Differentiation mistake (omitting the cubic term’s derivative factor).
• Assuming the zero shear location is always at mid-span; it depends on M(x).

Final Answer:
x = L/√3