Falling Chimney

A tall, slender, cylindrical brick chimney of height L is slightly perturbed from its vertical equilibrium position so that it topples over, rotating rigidly around its base B until it breaks at a point P. Show that the most likely value for the distance l of P from B is L/3. Assume that the chimney breaks because the torque is too great and the chimney bends and snaps (see Figure 1.1).

Figure 1.1

SOLUTION

First calculate the motion of the entire chimney of mass m by considering the torque τ about its base B (see Figure 1.2)

Figure 1.2

The moment of inertia about the base is I = (1/3)mL². The equation of motion for θ is found from

The piece of the chimney above the point P rotates in response to the torque τ_cm produced by its center of mass about P given by

and the torque τ(x) produced by the rest of the chimney attached below P, “trying” to convince the piece to rotate at

Find τ(x) by substituting for cosθ above:

Find the maximum torque by taking the derivative of τ(x) with respect to x and setting it equal to zero:

Either x = L, where the torque is zero, or L – x = (2/3)L → x = L/3, as was to be demonstrated.