Inside This Unit: The Full Breakdown
Simple Harmonic Motion (SHM) describes the back-and-forth oscillation of objects around an equilibrium position. This unit covers springs, pendulums, and the mathematical relationships between period, frequency, amplitude, and energy in oscillating systems.
Why it matters
SHM appears on every AP Physics 1 exam. You must understand the relationships between displacement, velocity, acceleration, and energy during oscillation. Spring and pendulum period formulas are frequently tested, and energy analysis of oscillating systems is a common free-response topic.
Key concepts
- Simple harmonic motion occurs when a restoring force is proportional to displacement: F = −kx (Hooke's law for springs).
- Period of a spring: T = 2π√(m/k). Period of a pendulum: T = 2π√(L/g). Period is independent of amplitude for SHM.
- In SHM, displacement, velocity, and acceleration are sinusoidal and out of phase: maximum speed occurs at equilibrium; maximum acceleration occurs at maximum displacement.
- Total energy in SHM is constant: energy oscillates between kinetic (½mv²) and potential (½kx²), with total E = ½kA² where A is amplitude.
Restoring Forces and Hooke's Law
Simple harmonic motion requires a restoring force that pushes the object back toward an equilibrium position with a magnitude proportional to the displacement. For a mass on a spring, Hooke's law gives this force: F = −kx, where k is the spring constant (stiffness) and x is the displacement from equilibrium. The negative sign indicates the force opposes the displacement. When released from a displaced position, the mass oscillates back and forth, passing through equilibrium with maximum speed and reversing direction at the extreme positions (amplitude, A). A simple pendulum approximates SHM for small angles (< 15°), where the restoring force is the tangential component of gravity. The motion repeats with a constant period regardless of amplitude — a property called isochronism.
Period, Frequency, and Motion Analysis
The period (T) is the time for one complete oscillation, and frequency (f = 1/T) is the number of oscillations per second, measured in hertz (Hz). For a mass-spring system, T = 2π√(m/k) — period depends on mass and spring constant but not amplitude. For a simple pendulum, T = 2π√(L/g) — period depends on length and gravitational acceleration but not mass or amplitude. During SHM, displacement, velocity, and acceleration vary sinusoidally with time. At the equilibrium position (x = 0), velocity is maximum and acceleration is zero. At the extreme positions (x = ±A), velocity is zero and acceleration is maximum. Acceleration is always directed opposite to displacement because the restoring force opposes the displacement.
Energy in Simple Harmonic Motion
The total mechanical energy of an ideal simple harmonic oscillator is constant and equals E = ½kA², where A is the amplitude. As the object oscillates, energy transforms continuously between kinetic energy (½mv², maximum at equilibrium) and potential energy (½kx², maximum at the extremes). At any position, KE + PE = ½kA². This means that at the equilibrium position, all energy is kinetic: ½mv²_max = ½kA², giving v_max = A√(k/m). At the amplitude, all energy is potential and velocity is zero. Energy diagrams showing KE and PE as functions of position are effective tools for understanding this oscillation. In real systems, friction or air resistance gradually converts mechanical energy into thermal energy, causing the amplitude to decrease over time (damped oscillation).
AP exam tip
For AP Physics 1 SHM problems, remember that period does NOT depend on amplitude. Students frequently confuse this. Also, know that maximum speed occurs at equilibrium (not at the endpoints) and maximum acceleration occurs at the endpoints (not at equilibrium).
Connections to other units
- Unit 2 (Dynamics): The restoring force (F = −kx) is analyzed using Newton's second law to derive the oscillation behavior.
- Unit 4 (Energy): Conservation of energy applied to springs and pendulums explains the continuous exchange between KE and PE.
- Unit 7 (Torque): A physical pendulum involves torque and rotational inertia, extending SHM to rotational systems.
