Simple Harmonic Motion Cheat Sheet

The core ideas of Simple Harmonic Motion distilled into a single, scannable reference — perfect for review or quick lookup.

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Quick Reference

Simple Harmonic Motion (SHM)

Oscillatory motion where the restoring force is proportional to displacement from equilibrium: F = -kx. The motion is sinusoidal with constant amplitude, period, and frequency.

Amplitude

The maximum displacement from the equilibrium position during oscillation. It determines the total energy of the system: E = (1/2)kA^2 for a spring system.

Period and Frequency

Period (T) is the time for one complete oscillation. Frequency (f) is the number of oscillations per second: f = 1/T. Angular frequency: omega = 2*pi*f = 2*pi/T.

Restoring Force

A force that always points toward the equilibrium position and is proportional to displacement. For a spring: F = -kx (Hooke's Law). This force is what causes the oscillation.

Spring Constant (k)

A measure of a spring's stiffness, defined as the force per unit displacement: k = F/x. A stiffer spring has a larger k and produces a higher-frequency oscillation for the same mass.

Energy in SHM

Total mechanical energy is constant and equals (1/2)kA^2. Energy oscillates between kinetic (max at equilibrium, KE = (1/2)mv^2) and potential (max at amplitude, PE = (1/2)kx^2). At any point: E = KE + PE.

Resonance

The phenomenon where the amplitude of a driven oscillation becomes maximum when the driving frequency equals the system's natural frequency. At resonance, energy transfer from the driver to the oscillator is most efficient.

Damped Oscillation

Oscillation in which the amplitude decreases over time due to energy loss from friction, air resistance, or other dissipative forces. The system eventually comes to rest at equilibrium.

Simple Pendulum

A point mass suspended by a massless string that oscillates under gravity. For small angles (less than about 15 degrees), it approximates SHM with period T = 2*pi*sqrt(L/g), independent of mass and amplitude.

Key Terms at a Glance

Simple Harmonic Motion:Oscillatory motion with restoring force proportional to displacement: F = -kx. Position follows x(t) = A*cos(omega*t + phi).

Amplitude:Maximum displacement from equilibrium. Determines total energy: E = (1/2)kA^2.

Period:Time for one complete oscillation cycle. T = 1/f = 2*pi/omega.

Frequency:Number of oscillations per second: f = 1/T. Measured in hertz (Hz).

Angular Frequency:omega = 2*pi*f = 2*pi/T. For a spring: omega = sqrt(k/m). For a pendulum: omega = sqrt(g/L).

Restoring Force:Force directed toward equilibrium, proportional to displacement. F = -kx for springs.

Spring Constant:Stiffness of a spring: k = F/x. Units: N/m. Larger k means stiffer spring and higher frequency.

Hooke's Law:F = -kx. The restoring force of a spring is proportional to displacement and opposite in direction.

Resonance:Maximum amplitude response when driving frequency matches natural frequency. Can be constructive or destructive.

Damped Oscillation:Oscillation with decreasing amplitude due to energy dissipation. Three regimes: underdamped, critically damped, overdamped.

Natural Frequency:The frequency at which a system oscillates when disturbed and allowed to move freely. f_0 = (1/2*pi)*sqrt(k/m).

Isochronism:The property that the period of SHM is independent of amplitude. Enables clocks and timing devices.

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