AP Physics 1 Unit 7: Simple Harmonic Motion (SHM) Foundations

Simple Harmonic Motion (SHM)

Oscillatory motion in which acceleration always points toward equilibrium and is proportional in magnitude to displacement: $a = -\theta^2x$.

Equilibrium Position

The position defined as x = 0 where, if the object is placed there and released, it remains at rest; restoring force is zero there.

Displacement (x)

Signed distance from equilibrium (x > 0 on one side, x < 0 on the other) used in SHM equations.

Restoring Force

A force that acts to return a system to equilibrium; in SHM it always points toward equilibrium.

Linear Restoring Force

A restoring force whose magnitude is directly proportional to displacement (double x → double force), producing SHM behavior.

SHM Acceleration Condition

Defining SHM relationship between acceleration and displacement: $a = -\theta^2x$ (opposite direction; proportional magnitude).

Angular Frequency (ω)

Constant that sets how fast SHM repeats; units rad/s; appears in $a = -\omega^2 x$ and relates to timing by $\omega = \frac{2\pi}{T} = 2\pi f$.

Negative Sign in SHM Equations

Indicates the acceleration/force is opposite the displacement (toward equilibrium), not that acceleration is always negative.

Hooke’s Law

Ideal spring force model: F = −kx, where k is the spring constant and the force is restoring and linear.

Spring Constant (k)

Measure of spring stiffness in Hooke’s law; units N/m; equals the magnitude of the slope of an F vs. x graph.

Newton’s Second Law (in SHM context)

Using $ΣF = ma$ with a linear restoring force ($-kx$) leads to SHM: $a = -\frac{k}{m}x$.

Mass–Spring Angular Frequency

For an ideal mass–spring system, $ω = \sqrt\frac{k}{m}$, found by comparing $a = -\frac{k}{m}x$ to $a = -\theta^2x$.

Mass–Spring Oscillator

A mass attached to an ideal spring (no friction, obeys Hooke’s law) that exhibits SHM.

Simple Pendulum

A mass (bob) on a light string of length L swinging under gravity; approximately SHM only for small angles.

Small-Angle Approximation

For a pendulum at small angles (in radians), sinθ ≈ θ, making the restoring effect proportional to displacement and enabling SHM modeling.

Non-SHM Oscillation

Back-and-forth motion that does NOT satisfy proportional-and-opposite restoring acceleration/force (e.g., nonlinear restoring force or bouncing ball).

Force–Displacement Graph (F vs. x) in SHM

A straight line through the origin with slope −k for an ideal spring; linearity is a signature of SHM.

Acceleration–Displacement Graph (a vs. x) in SHM

A straight line through the origin with slope $-\theta^2$; the slope magnitude gives $\theta^2$.

Turning Points (Maximum Displacement)

Locations where |x| is maximum, speed v = 0, and |a| (and restoring force magnitude) is maximum.

Equilibrium Crossing (x = 0)

Point where acceleration a = 0 (and restoring force is 0) while speed magnitude is maximum.

Period (T)

Time for one complete cycle of repeating motion; measured in seconds (s).

Frequency (f)

Number of cycles per second; measured in hertz (Hz), where $1 \text{ Hz} = 1 \text{ s}^{-1}$; related by $f = \frac{1}{T}$.

Angular Frequency Relationships

Connections among timing quantities: ω = 2πf and ω = 2π/T.

Mass–Spring Period Formula

For an ideal mass–spring oscillator: $T = 2\pi\sqrt\frac{m}{k}$ and $f = \frac{1}{2\pi}\sqrt\frac{k}{m}$.

Simple Pendulum Period Formula (Small Angle)

For a small-angle pendulum: $T = 2\pi\sqrt\frac{L}{g}$ and $f = \frac{1}{2\pi}\sqrt\frac{g}{L}$; independent of bob mass in the model.