Oscillations – AP Physics 1

Simple Harmonic Motion (SHM)

Periodic motion where the restoring force is proportional to and opposite in direction to displacement from equilibrium.

Restoring Force

A force that acts to bring an object back toward its equilibrium position in SHM.

Hooke’s Law

F = −kx the restoring force in a spring is proportional to the displacement.

Spring Constant (k)

A measure of a spring’s stiffness

Displacement (x)

The distance from the equilibrium position in SHM.

Amplitude (A)

The maximum displacement from the equilibrium point.

Period (T)

Time taken to complete one full oscillation; measured in seconds.

Frequency (f)

Number of oscillations per second; measured in hertz (Hz).

Relationship between Period and Frequency

f = 1T

Angular Frequency (ω)

Rate of oscillation in radians/sec; ω = 2πf

Mass-Spring System Period

T = 2π √m​​​​/k

Simple Pendulum Period

T = 2π √L/g ​​

Equilibrium Position

The position where the net force on the object is zero in SHM.

Velocity in SHM

v(t) = −Aω sin⁡(ωt+ϕ)

Acceleration in SHM

a(t) = −Aω²cos⁡(ωt+ϕ)

Phase Constant (φ)

Determines the initial position and direction of motion in SHM.

Kinetic Energy in SHM

KE = ½ mv² maximum at equilibrium position.

Potential Energy in SHM (spring)

PE = ½ kx² maximum at maximum displacement.

Total Mechanical Energy in SHM

Constant; E = KE + PE = ½ kA²

Displacement-Time Graph

A cosine or sine wave showing periodic displacement.

Velocity-Time Graph

A sine or cosine wave shifted 90° out of phase from displacement.

Acceleration-Time Graph

A cosine wave 180° out of phase with displacement.

SHM Energy Transfer

Energy continuously transfers between kinetic and potential forms.

Conditions for SHM

Restoring force must be proportional to displacement and directed toward equilibrium.

Small-Angle Approximation

For pendulums SHM applies when angles are

Damping

A force (like friction or air resistance) that reduces amplitude over time.

Resonance

Occurs when an external force matches the natural frequency of a system

Natural Frequency

The frequency at which a system oscillates without external driving forces.

Real-World Example of SHM

A child swinging on a swing (simple pendulum).

Energy Conservation in SHM

Total mechanical energy remains constant in the absence of damping.