Introduction to Classical Mechanics · Problem 3.60

Oscillations — Morin

Physics — Mechanics · circular-pendulum

Problem

Circular and plane pendulums * Consider the circular pendulum in the example in Section 3.5. Let the

x

y

plane be the horizontal plane of the circle. For small

\beta

, what (approximately) is the

F_x

component of the force on the mass when it is at the position

(x, y)

on the circle? Consider now a standard plane pendulum where the mass swings back and forth in the vertical plane containing the

x

axis. Let the maximum angle of the pendulum be the same small angle

\beta

. What (approximately) is the

F_x

component of the force on the mass in terms of its

x

coordinate? Your two results should be the same, which means that the

x

motions of the two systems are the same, because when they are each at their maximum

x

value (

\ell \sin\beta

), they have the same

x

speed (zero), and we just showed that they always have the same

x

acceleration, independent of any

y

motion. So the frequencies of the two systems must be equal. (We'll see in Section 4.2 that the frequency of a plane pendulum is

\sqrt{g/\ell}

, in agreement with this observation.)