11.2. The Pendulum

Galileo’s Discovery

One Sunday in 1583, as Galileo Galilei attended Mass in a cathedral in Pisa, Italy, he noticed something interesting about the chandeliers hanging from the ceiling, which were swaying slightly due to air currents. The chandeliers hanging from equal distances had the same period, even if they had different masses, while chandeliers hanging from different distances had different periods, even if the chandeliers were otherwise identical.

The Duomo of Santa Maria Assunta in

Pisa, Italy

Galileo at seventeen

Simple Pendulums

At rest, the bob hangs straight down in its equilibrium position. When displaced by a small angle and released, the bob oscillates from side to side.

Pendulums move with Simple Harmonic Motion

A pendulum moves with simple harmonic motion. On a paper strip moving horizontally, it would trace out a sine wave.

Pendulums with Simple Harmonic Motion

Pendulums with Simple Harmonic Motion

Example 1

Example 2

Why doesn’t a pendulum’s period depend on mass?

Why doesn’t a pendulum’s period depend on amplitude?

Why does a pendulum’s period depend on length?

Example 3

Damped Harmonic Motion

The amplitude of an oscillating spring or pendulum actually decreases, or dampens, over time due loss of kinetic energy to friction (internal and external) and air resistance.

Damped Harmonic Motion

However, if the damping is very large, it will dampen on the first oscillation.

A: underdamping: there are a few oscillations before the oscillator comes to rest.

C: overdamping: the system is slowed so much by friction that it takes a long time to get back to equilibrium.

B: critical damping: this amount of damping yields the fastest way to get to back to equilibrium. This is the ideal for shock absorbers, swinging doors, etc.

Forced Harmonic Motion

Resonance

For example, pushing a child on a swing at the pendulum’s resonant frequency increases their amplitude with the least amount of work.

Soldiers marching across a bridge are generally told to break step to avoid resonant frequencies.

The Tacoma Narrows Bridge began to resonate when the winds reached 40 mph.

[see video]

Resonance

Bridges and tall buildings are now designed with damping systems to avoid oscillations and resonance.

Dampers placed inside a building for

earthquake protection.

The Pendulum Clock

Resonance may be desirable or undesirable.

For centuries the most precise timepiece was the pendulum

clock. A weight or spring supplied potential energy at

the resonant frequency interval to compensate for kinetic energy loss due to

friction and air drag. This kept the amplitude constant.

Solid Pendulum Arms

The Song of the Spider

The next time you see a spider’s web, move close and start to hum, starting at a low pitch. Slowly increase the pitch of your humming until you see the spider react. The web will vibrate at its resonant frequency, and the spider will think it just snagged lunch.