AP Physics C Mechanics Unit 6 Notes: Learning Simple Harmonic Motion

Simple Harmonic Motion (SHM)

Back-and-forth motion in which acceleration is proportional to displacement from equilibrium and directed toward equilibrium: a = −ω²x.

Equilibrium Position

The position where the net force (or net torque) is zero; SHM displacement x is measured from this point.

Displacement (x) from Equilibrium

The signed position measured relative to equilibrium; in SHM it determines the restoring force/acceleration (e.g., a ∝ −x).

Restoring Force

A force that acts to return a system to equilibrium; for SHM it must be (approximately) proportional to displacement and opposite in direction.

Angular Frequency (ω)

A constant (rad/s) that sets the time scale of SHM; relates to acceleration by a = −ω²x and to period by ω = 2π/T.

Hooke’s Law

Spring force is proportional to displacement: F_s = −kx (valid when the spring is not stretched/compressed too far).

Spring Constant (k)

A measure of spring stiffness (N/m) appearing in Hooke’s law; larger k means a “stiffer” spring.

Mass–Spring Oscillator (Horizontal)

A mass m attached to a spring on a frictionless surface that undergoes SHM with ω = √(k/m).

Vertical Spring Equilibrium Extension (x_eq)

The stretch where the hanging mass is at rest: kx_eq = mg; oscillations occur about this shifted equilibrium.

Simple Pendulum

A point mass (bob) on a massless string of length L; for small angles it approximates SHM with ω = √(g/L).

Tangential Component of Gravity (Pendulum)

The restoring force along the arc: F_t = −mg sinθ, pointing toward θ = 0.

Torque (τ) About a Pivot

Rotational analog of force that causes angular acceleration; for a pendulum τ = −mgL sinθ.

Moment of Inertia (I) for a Point Mass Pendulum

For a bob of mass m at distance L from the pivot: I = mL².

Angular Acceleration (α)

The second time derivative of angle: α = d²θ/dt²; related to torque by Στ = Iα.

Small-Angle Approximation

For sufficiently small θ (in radians), sinθ ≈ θ, allowing the pendulum equation to become SHM: θ'' + (g/L)θ = 0.

Period (T)

Time for one full oscillation (seconds); T = 1/f and T = 2π/ω.

Frequency (f)

Number of cycles per second (Hz); f = 1/T and f = ω/(2π).

Hertz (Hz)

Unit of frequency meaning s⁻¹ (cycles per second).

Phase Constant (φ)

A constant that shifts the SHM graph in time; set by initial conditions in x(t) = A cos(ωt + φ).

Amplitude (A)

Maximum displacement from equilibrium; sets maximum speed/energy but (for ideal SHM) does not change the period.

SHM Differential Equation

The defining equation of motion: x'' + ω²x = 0 (or θ'' + ω²θ = 0 for small-angle pendulum).

Sinusoidal Solution to SHM

A solution of the form x(t) = A cos(ωt + φ) (equivalently sine), which satisfies x'' = −ω²x.

Maximum Speed (v_max) in SHM

Occurs at equilibrium; vmax = Aω (also from energy: ½kA² = ½mvmax² for a spring).

Maximum Acceleration (a_max) in SHM

Occurs at maximum displacement; a_max = Aω² (since a = −ω²x).

Mechanical Energy Conservation in Ideal SHM

With no nonconservative forces, total energy stays constant and shifts between kinetic and potential (spring: E = ½mv² + ½kx²).