7.1 Defining Simple Harmonic Motion (SHM)
Not all repetitive motion is Simple Harmonic Motion (SHM). For a system to undergo SHM, there must be a specific type of restoring force acting on the object.
Simple Harmonic Motion: Oscillatory motion under a retarding force proportional to the amount of displacement from an equilibrium position.
The classic example is a mass on a spring governed by Hooke's Law (Fs = -kx). Because the force is always pulling opposite the displacement, it constantly tries to pull the mass back to equilibrium, causing it to overshoot and oscillate.
7.2 Frequency and Period of SHM
How fast does an oscillator vibrate? We describe the timing of cycles using Period and Frequency.
Period (T)
The time required to complete one full cycle of oscillation.
Units: Seconds (s)
Frequency (f)
The number of full cycles completed in one second.
Units: Hertz (Hz) or s-1
Relationship: f = 1 / T
The Period Equations
The time it takes for a system to oscillate depends entirely on its physical properties. Amplitude NEVER affects the period of an ideal SHM system!
| Mass-Spring System | Simple Pendulum |
|---|---|
|
Ts = 2π
√
m
k
|
Tp = 2π
√
L
g
|
|
|
7.3 Representing and Analyzing SHM
Because the restoring force changes with position, the acceleration is not constant. This means we cannot use the Big Three kinematics equations from Unit 1! Instead, the motion traces out sinusoidal waves (sine and cosine).
The Dance of x, v, and a
At the Maximum Displacement (Amplitude, ±A):
- Velocity is Zero (it must momentarily stop to turn around).
- Restoring force is Maximum (spring is fully stretched/compressed).
- Acceleration is Maximum (since F=ma).
At Equilibrium (x = 0):
- Restoring force is Zero (spring is relaxed).
- Acceleration is Zero.
- Velocity is at its Maximum (it built up speed falling toward the center).
7.4 Energy of Simple Harmonic Oscillators
For an ideal oscillator with no friction or air resistance, the Total Mechanical Energy (E) of the system remains perfectly conserved. It continually sloshes back and forth between Potential Energy (U) and Kinetic Energy (K).
🎯 Reading the Energy Graph:
Notice that at the amplitudes (±A), the Potential Energy curve hits the Total Energy line, meaning K=0. At equilibrium (x=0), the Kinetic Energy curve hits the Total Energy line, meaning U=0.
Unit 7 Key Takeaways
SHM requires a restoring force proportional to displacement (F ∝ −x).
Amplitude does NOT affect the period of an ideal spring or pendulum.
A stiffer spring (high k) oscillates faster (smaller T). A heavier mass oscillates slower (larger T).
Acceleration is maximum at the amplitude and zero at equilibrium.
Velocity is zero at the amplitude and maximum at equilibrium.
Total energy is constant. U is max at amplitudes; K is max at equilibrium.
End of Unit 7 Study Guide. Good luck on your AP Exam!