Simple Harmonic Motion: every key term you need (+ practice quiz)

More AP Physics 1 guides

• Kinematics
• Forces and Newton's Laws
• Work, Energy and Power
• Momentum and Collisions
• Circular Motion and Gravitation

Simple Harmonic Motion (SHM)

Oscillation about equilibrium; restoring force proportional to displacement. a = -ω²x.

Restoring Force

Force toward equilibrium. F = -kx (spring). Proportional to displacement but opposite direction.

Spring Constant (k)

Stiffness of spring. F = -kx. Stiffer spring = larger k. Units: N/m.

Period of Spring

T = 2π√(m/k). Heavier mass → longer period. Stiffer spring → shorter period.

Pendulum Period

T = 2π√(L/g). Depends on length, gravity. Independent of mass/amplitude (small angle).

Energy in SHM

Total = kinetic + potential = constant. At equilibrium: max KE, min PE. At extremes: min KE, max PE.

Amplitude

Maximum displacement from equilibrium. Determines energy (E ∝ A²). All oscillations same frequency, different amplitudes.

Frequency & Period

Frequency f = 1/T. Angular frequency ω = 2πf = 2π/T. SHM frequency independent of amplitude.

Equations of SHM

x(t) = A·cos(ωt + φ). v(t) = -Aω·sin(ωt + φ). a(t) = -Aω²·cos(ωt + φ).

Damping

Friction reduces amplitude over time. Underdamped: oscillates (slow decay). Overdamped: no oscillation. Critical: returns slowly to equilibrium.

Unit 6 Summary

SHM: oscillation with F ∝ displacement. Period depends on mass/stiffness (spring) or length/gravity (pendulum). Energy transforms between KE and PE.

Phase Relationships

In SHM, velocity is maximum where displacement is zero and acceleration is maximum (and opposite in sign) where displacement is maximum. x, v, a graphs are shifted quarter-periods apart.

Maximum Speed and Acceleration

v_max = Aω = A√(k/m) and a_max = Aω² = Ak/m. Doubling amplitude doubles both, but does not change the period.

Amplitude Independence of Period

For an ideal spring-mass system, T = 2π√(m/k) with no dependence on amplitude. A pendulum shares this property only for small angles.

Vertical Spring Equilibrium

A mass hanging on a spring stretches it by mg/k to a new equilibrium; oscillations about that point have the same period as horizontal SHM. Gravity shifts the center but not the frequency.

Springs in Series and Parallel

Parallel springs (side by side) add: k_eff = k₁ + k₂, stiffer. Series springs (end to end): 1/k_eff = 1/k₁ + 1/k₂, softer. Cutting a spring in half doubles its constant.

Energy Split in SHM

At x = A/2, potential energy is ¼ of total and kinetic is ¾. KE equals PE when x = A/√2 ≈ 0.71A, not at half amplitude.

Pendulum on Another Planet

T = 2π√(L/g), so a pendulum clock runs slow where g is smaller (Moon: period √6 ≈ 2.45 times longer). Spring clocks are unaffected by g.

Pendulum in an Accelerating Elevator

Effective g becomes g ± a. Accelerating upward shortens the period; in free fall the pendulum does not oscillate at all.

Restoring Force Requirement

SHM requires a force proportional to and opposite displacement (F = −kx). A bouncing ball has a constant-magnitude force (gravity), so it is periodic but not SHM.

Frequency from a Graph

Read the period as the time between successive identical points on an x-t curve (e.g., peak to peak). Frequency is 1/T; angular frequency is 2π/T.

Timing Different Segments

Because motion is fastest through equilibrium, the mass spends only T/12 traveling from x = 0 to A/2 but T/6 traveling from A/2 to A. Time is not proportional to distance in SHM.

Mass on Spring vs Pendulum Mass Dependence

Adding mass lengthens a spring's period (T ∝ √m) but leaves a simple pendulum's period unchanged, because for the pendulum both restoring force and inertia scale with m.

Physical Pendulum (qualitative)

A rigid swinging object like a meter stick pivoted at one end has a period different from a simple pendulum of the same length; the mass distribution matters. AP 1 treats this qualitatively.

Resonance and Damping

Driving a system at its natural frequency produces large-amplitude oscillation. Damping removes energy each cycle, decreasing amplitude while barely changing the period.