7.3 Sinusoidal Relationships in Simple Harmonic Motion

Position, velocity, acceleration equations

For simple harmonic motion (SHM), position is x(t) = A cos(ωt + φ), where A is amplitude, ω is angular frequency, and φ is the phase constant. Velocity is the time derivative: v(t) = -Aω sin(ωt + φ), and acceleration is a(t) = -Aω² cos(ωt + φ) = -ω²x. Acceleration is always proportional to and opposite in sign from displacement.

Amplitude, period, and angular frequency

Amplitude (A) is the maximum displacement from equilibrium; it doesn't affect period or frequency. Angular frequency ω = 2π/T = 2πf sets how fast the phase advances. Maximum speed is v_max = Aω, occurring at x = 0; maximum acceleration is a_max = Aω², occurring at maximum displacement, where restoring force is greatest.

Phase relationships and energy

Velocity leads position by 90° (π/2), and acceleration is exactly out of phase (180°) with position. When |x| is max, v = 0 and a is max; when x = 0, v is max and a = 0. Total mechanical energy remains constant, continuously exchanging between kinetic energy (max at x=0) and potential energy (max at extremes).

Key terms

x(t) for SHM

x(t) = A cos(ωt + φ), where A is amplitude, ω is angular frequency, φ is the phase constant.

v(t) for SHM

v(t) = -Aω sin(ωt + φ), the derivative of position; maximum magnitude is Aω at x = 0.

a(t) for SHM

a(t) = -Aω² cos(ωt + φ) = -ω²x(t); maximum magnitude Aω² occurs at maximum displacement.

Phase relationships in SHM

Velocity leads position by 90°; acceleration is 180° out of phase with position (opposite direction).

Energy exchange in SHM

Kinetic energy is maximum at x = 0 (max speed); potential energy is maximum at extremes (x = ±A); total mechanical energy is constant.

Try an AP-style question

x(t) = A cos(ωt). What is v(t)?

1. −Aω sin(ωt)
2. −Aω² cos(ωt)
3. Aω sin(ωt)
4. A cos(ωt)/ω