Hold your pencil horizontally by its tip. To keep it still, you will have to exert a combination of forces on its bottom and top. These forces can be viewed as a superposition of a net upward force, and a pair of opposite forces. The former ensures the forces on the pencil are balanced, while the latter provides a torque, called the bending moment, which ensures the torques on the pencil are balanced. Since the bending moment arises from a pair of opposite forces, it doesn't depend on the choice of origin.
a. Consider a rod of length ℓ and mass per length λ. What is the bending moment you must exert to hold the rod horizontally at its end?
Just as each piece of a string exerts a tension on neighboring pieces in equilibrium, each piece of a solid rod also exerts a bending moment on its neighbors. For thin rods under heavy loads, this bending moment can be the limiting factor that causes them to break.
[Figure: Two diagrams showing rods supported at both ends. Left diagram shows supports at both ends of length ℓ. Right diagram shows supports at left end and distance d from left end, with d > ℓ/2.]
Suppose a rod is supported at both ends, so that it forms a bridge, as shown at left above. Assume the supports are simple, so that they only provide an upward force, and no bending moment. In equilibrium, a bending moment will appear throughout the rod. The magnitude of the maximum bending moment the rod can exert at any point without breaking is M₀, and the length of the rod is ℓ. The bridge is loaded uniformly, with a mass per length of λ (including its own mass).
b. Find the maximum possible value of λ before the bridge collapses.
c. Now suppose that one support remains at the left end, while the other is a distance d > ℓ/2 away from the left end, as shown at right above. In static equilibrium, find the bending moment M(x₀) at a distance x₀ < d from the left end.
d. Find the value of d that maximizes the load λ that the bridge can take before collapsing.