AP Physics 1 Unit 7 Oscillations: Understanding SHM Through Representations and Energy

Simple Harmonic Motion (SHM)

Oscillatory motion in which the acceleration (and net force) is proportional to displacement from equilibrium and opposite in direction.

Restoring force

A force that points toward equilibrium and tends to return an object to its equilibrium position.

SHM defining equation

The condition for SHM: $a = -\omega^2 x$ (acceleration is proportional to displacement and opposite in direction).

Hooke’s Law

Spring restoring-force relationship: F = −kx, where k is the spring constant and x is displacement from equilibrium.

Spring constant (k)

A measure of a spring’s stiffness (units N/m) in Hooke’s law $F = -kx$.

Angular frequency (ω)

A measure of how fast an oscillator cycles, in radians per second; for a mass-spring system $\theta = \frac{\sqrt{k}}{m}$.

Equilibrium position

The position where net force is zero (often x = 0 in SHM).

Amplitude (A)

The maximum displacement from equilibrium in an oscillation.

Period (T)

The time required for one complete cycle of oscillation.

Frequency (f)

The number of cycles per second; $f = \frac{1}{T}$.

Angular frequency relations

Connections among period and frequency: ω = 2πf = 2π/T.

Mass–spring period

For an ideal mass–spring oscillator: T = 2π√(m/k), independent of amplitude A.

Small-angle pendulum period

For a simple pendulum at small angles: $T = 2\theta \text{π√}\frac{L}{g}$, where L is length and g is gravitational field strength.

Small-angle approximation

For small angles (in radians), sinθ ≈ θ, allowing pendulum motion to be modeled as SHM.

Effective spring constant of a small-angle pendulum (k_eff)

For small oscillations, the pendulum’s tangential restoring force can be written like Hooke’s law with $k_{eff} = \frac{mg}{L}$.

Sinusoidal position model

Standard SHM position function: x(t) = A cos(ωt + φ).

Phase constant (φ)

A constant that sets the oscillator’s starting point in the cycle at t = 0 (horizontal shift of the sinusoid).

SHM velocity function

For x(t) = A cos(ωt + φ), the velocity is v(t) = −Aω sin(ωt + φ).

SHM acceleration function

For x(t) = A cos(ωt + φ), the acceleration is a(t) = −Aω² cos(ωt + φ).

Maximum speed in SHM (v_max)

The greatest speed occurs at equilibrium: $v_{max} = A \times \theta$.

Maximum acceleration magnitude in SHM (a_max)

Acceleration magnitude is greatest at the endpoints: $a_{max} = A \times \theta^2$.

a vs. x “fingerprint” of SHM

In SHM, an acceleration–displacement graph is a straight line through the origin with slope a/x = −ω².

Elastic potential energy (spring)

Energy stored in a spring: $U_s = \frac{1}{2} kx^2$, where $x$ is instantaneous displacement.

Total mechanical energy (ideal spring SHM)

Constant (if no damping): $E = \frac{1}{2}mv^2 + \frac{1}{2}kx^2 = \frac{1}{2}kA^2 = \frac{1}{2}m\theta^2A^2$.

Speed as a function of displacement (spring SHM)

From energy conservation: $v = \sqrt\frac{k}{m}(A^2 - x^2)$ (speed depends on position; $v = 0$ at $x = ±A$).