AP Physics 1 Unit 7 Study Notes: Oscillations and Simple Harmonic Motion

Oscillation

Repeated back-and-forth motion about an equilibrium position.

Equilibrium (in oscillations)

The special position where the net force on the object is zero; if placed there at rest, it stays there.

Restoring force

A force that points toward equilibrium whenever the object is displaced from equilibrium.

Stable equilibrium

An equilibrium where a small displacement produces a restoring force back toward equilibrium (required for oscillations).

Unstable equilibrium

An equilibrium where a small displacement produces a force away from equilibrium, so the system “runs away” rather than returning.

Period (T)

Time for one complete cycle of oscillation (units: seconds).

Frequency (f)

Number of oscillation cycles per second (units: hertz, Hz).

Frequency–period relationship

Frequency and period are reciprocals: f = 1/T.

Angular frequency (ω)

A measure of how fast the oscillation phase advances (units: rad/s).

Angular frequency–frequency relationship

ω = 2πf.

Angular frequency–period relationship

ω = 2π/T.

Amplitude (A)

Maximum displacement from equilibrium during oscillation.

Phase

A label for where the oscillator is within its cycle (its “place” in the oscillation).

Phase constant (φ)

A constant that shifts a sinusoidal SHM function left/right in time to match initial conditions.

Simple harmonic motion (SHM)

Oscillation where the restoring force is proportional to displacement and opposite in direction (restoring).

Hooke’s law

For an ideal spring, the spring force is restoring and proportional to displacement: F_s = −kx.

Spring constant (k)

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

Displacement (x) in SHM

Signed distance from the equilibrium position (often defined so x = 0 at equilibrium).

Natural (unstretched) length

The length of a spring when no external forces stretch or compress it.

Equilibrium position (net force zero)

The position where all forces balance so net force is zero (for a vertical spring, this is below the natural length).

SHM acceleration relation (spring)

From F = −kx and F = ma: a = −(k/m)x (acceleration proportional to x and opposite in sign).

Turning point

A location of maximum displacement (x = ±A) where the object reverses direction and v = 0 momentarily.

Equilibrium crossing behavior

At x = 0, the restoring force is zero so a = 0, and the speed is maximum.

Sinusoidal position model for SHM

A standard SHM model is x(t) = A cos(ωt + φ) (or equivalently a sine form).

Mass–spring angular frequency

For an ideal mass-spring oscillator, ω = √(k/m).

Mass–spring period

For an ideal mass-spring oscillator, T = 2π√(m/k).

Amplitude independence of period (ideal spring SHM)

In an ideal mass-spring SHM system, the period T does not depend on amplitude A.

Vertical spring equilibrium condition

For a hanging mass at rest: kΔL = mg (spring force balances weight).

Static stretch (ΔL)

The equilibrium extension of a vertical spring: ΔL = mg/k.

Equilibrium shift in vertical springs

Gravity shifts the equilibrium position, but oscillations are still SHM about the shifted equilibrium (and the period is unchanged for an ideal spring).

Spring potential energy (U_s)

Energy stored in an ideal spring: U_s = (1/2)kx^2 (depends on x^2, same for +x and −x).

Kinetic energy (K)

Energy of motion: K = (1/2)mv^2.

Total mechanical energy in spring SHM

For an ideal mass-spring oscillator with amplitude A: E = (1/2)kA^2 (constant if no damping).

Energy exchange in SHM

In ideal SHM, energy continuously trades between potential energy (max at turning points) and kinetic energy (max at equilibrium) while total E stays constant.

Speed–position relation in SHM

Using energy: v = √[(k/m)(A^2 − x^2)] = ω√(A^2 − x^2).

Maximum speed (where it occurs)

Speed is maximum at equilibrium (x = 0) in SHM.

Maximum acceleration magnitude (where it occurs)

In SHM, |a| is largest at maximum displacement (x = ±A), not at equilibrium.

Acceleration–position link (SHM)

A general SHM relation: a = −ω^2 x (acceleration is opposite in sign to position).

Simple pendulum

A small mass on a light string of length L that can oscillate; for small angles its motion is approximately SHM.

Tangential restoring force for a pendulum

Component of gravity along the arc: F_t = mg sinθ, directed toward equilibrium.

Small-angle approximation

For small angles (in radians), sinθ ≈ θ, making the pendulum’s restoring effect proportional to θ (enabling SHM modeling).

Pendulum period (small-angle)

For a simple pendulum at small angles: T = 2π√(L/g).

Pendulum period independence of mass

In the simple pendulum model (small angles), the period does not depend on the bob’s mass.

Damping

Energy loss due to nonconservative forces (friction/drag), causing total mechanical energy and amplitude to decrease over time.

Driven oscillator

An oscillator acted on by an external periodic force (a “driver”) that can add energy to the motion.

Natural frequency

The frequency an oscillator would have if left to oscillate on its own (without an external driving force).

Resonance

Large-amplitude response when the driving frequency is near the system’s natural frequency (efficient energy transfer).

Measuring period by timing many cycles

To reduce reaction-time error, time N cycles and compute T = t_N / N.

Linearization to find spring constant k

From T = 2π√(m/k): T^2 = (4π^2/k)m. A plot of T^2 vs m has slope = 4π^2/k, so k = 4π^2/slope.

Linearization to find g (pendulum)

From T = 2π√(L/g): T^2 = (4π^2/g)L. A plot of T^2 vs L has slope = 4π^2/g, so g = 4π^2/slope.