The bending moment of a cantilever beam of length $$l$$ and carrying a uniformly distributed load of w per unit length is __________ at the free end.

1. Identify Beam Boundary Conditions.
A cantilever beam is fixed at one end and completely unsupported (free) at the other end. Because the free end has no external reaction forces or concentrated moments acting directly on its boundary, both the shear force and the bending moment must drop to zero at this point to maintain static equilibrium.
2. Formulate Bending Moment Equation
Let the free end be the origin x = 0 and progress toward the fixed end at x = l. The bending moment M(x) at any distance x from the free end is calculated by integrating the uniformly distributed load w: M(x)=-(w.x^2)/2
3. Evaluate at Free End
To find the bending moment specifically at the free end, substitute (x = 0) into the moment equation:
M(0)=-(w.0^2)/2=0
At the fixed end ((x = l)), the bending moment reaches its maximum absolute value of:
Mmax = -(w.l^2)/2