1. AP Physics Sections 11-4 to 11-6 Simple pendulum,
damping, and resonance

2. SHM in a pendulum
We looked at the pendulum in section 6.5 when discussing the conservation of mechanical energy.
A pendulum exhibits simple harmonic motion.
For small angles (less than 10 or 15 degrees), the equation for the period is given by:
T = 2π ℓ g
equilibrium position

3. Frequency of a simple pendulum
Determining that a simple pendulum is undergoing simple harmonic motion, and deriving the equation for the period of the pendulum, is left for you to do as one part of your formal lab report’s theory section.
Frequency of a simple pendulum 1
f =
Since
, frequency equals the reciprocal T
of the last equation for small angles. g 1
f = 2π ℓ

4. Mechanical clocks
The earliest mechanical clocks probably appeared in the later 13th century, but we know of examples from the 14th. They used falling weights to power a special gear called an escapement that could only move one way. Periodic motion was obtained by a swinging pair of masses on a rod (an early balance wheel) called the foliot. These clocks were placed in a town’s church or hall tower.

5. Pendulum clocks
Based on Galileo’s discovery that the period of a pendulum is independent of the amplitude, Christian Huygens invented the first pendulum clock in 1656.
He simply replaced the foliot balance rod with a swinging pendulum. The precision improved from an error of as much as half an hour a day to only minutes a day.

6. Huygens continued to improve his clock designs
Huygens continued to improve his clock designs. The pendulum clock remained the most precise timepiece for 270 years, until the invention of the quartz clock in 1927.
Grasshopper
escapement
John Harrison
1722
Huygens’ H6 clock
Gridiron pendulum
John Harrison, 1726

7. Damped harmonic motion example of light damping
Real oscillating systems like springs experience damping forces such as air resistance and internal friction. This creates damped harmonic motion.
The amplitude decreases according to an exponential function shown by the dashed lines.
The period and frequency change little except with very heavy damping.
example of light damping

8. If just the right amount of damping force is applied, the oscillator will come to rest in a minimum amount of time. This system is said to be critically damped.
If the damping force causes the oscillator to return to equilibrium without periodic motion, but does so over a long time, it is overdamped.
If the damping force allows the oscillator to exhibit some periodic motion it is said to be underdamped.
underdamped spring

9. Resonance
Recall that for a mass on a particular spring, the period only depends on mass, and for a pendulum it only depends on length.
If a force is periodically applied to the object in SHM, and if that period nearly equals the period of the system, the amplitude of the vibration will increase. This is resonance.
Even a small force in sync with the natural vibration period of the system can cause huge increases in amplitude.

10. The natural frequency of the oscillator is derived from the period equations:g 1 k 1
f0 =
f0 = 2π 2π m ℓ
natural frequency of
mass–spring system
natural frequency of
pendulum
If an outside periodic (vibrational) force is applied, it can change the amplitude. The biggest amplitude will result when f = f0.

11. Resonance occurs when f ≅ f0
Resonance occurs when f ≅ f0. Vibrations can be created that resonate with one oscillator.
The greatest amplitude will be seen in the oscillator with the same natural frequency as the one forcing the vibrations. f = f0
Resonance can cause vibrations so large than the oscillator can break. In one video example, a wine glass is broken with a resonant frequency of sound.
The 1940 Tacoma Narrows bridge in Washington collapsed due to catastrophic failure caused by resonance. Even on opening day, people knew there was a problem.