Question

A heavy round log is suspended at the ends on two ropes so that the distance between the points of suspension of the ropes is equal to the diameter of the log. The length of each vertical segment of the ropes is l.

Determine the period T of small oscillations of the system in a vertical plane perpendicular to the log.

Answer 1

Let us mark the horizontal diameter AB = 2r of the log at the moment it passes through the equilibrium position.

Let us now consider the log at the instant when the ropes on which it is suspended are deflected from the vertical by a small angle α (Fig. 187). 

In the absence of slippage of the ropes, we can easily find from geometrical considerations that the diameter AB always remains horizontal in the process of oscillations. Indeed, if EF ⊥ DK, FK = 2r tan α ≈ 2rα. But BD ≈ FK/2 ≈ rα. Consequently,   ∠BOD, α as was indicated above. Since the diameter AB remains horizontal all the time, the log performs translatory motion, i.e. the velocities of all its points are the same at each instant. Therefore, the motion of the log is synchronous to the oscillation of a simple pendulum of length I. Therefore, the period of small oscillations of the log is