What is simple harmonic motion (SHM)?

Simple harmonic motion is a type of periodic oscillation where the restoring force is directly proportional to the displacement from equilibrium. Examples include a mass on a spring and a pendulum swinging at small angles. The motion follows a sinusoidal pattern: x(t) = A cos(wt + phi).

How does pendulum length affect its period?

The period of a simple pendulum is T = 2*pi*sqrt(L/g), where L is the length and g is gravitational acceleration. Longer pendulums swing more slowly (larger period), while shorter pendulums swing faster. Notably, the period does not depend on the mass of the bob for small angles.

What happens to energy during simple harmonic motion?

In undamped SHM, total mechanical energy is conserved. Energy continuously converts between kinetic energy (KE = 1/2 mv^2, maximum at equilibrium) and potential energy (PE = 1/2 kx^2 for springs, maximum at extremes). The total energy E = 1/2 kA^2 remains constant. With damping, total energy gradually decreases over time.

Pendulum & SHM Simulator

Explore simple harmonic motion with an interactive pendulum and spring-mass system. Observe how length, mass, damping, and gravity affect period, frequency, and energy in real time.

Controls

Mode

Length (m)

1.0

Spring Constant k (N/m)

Mass (kg)

1.0

Amplitude (°)

Damping

0.0

Gravity

Presets

Values

Mode	Pendulum
Parameter	L = 1.0 m
Period (T)	—
Frequency (f)	—
Amplitude	30°
Current θ/x	—
Current ω/v	—
KE	—
PE	—

The Physics of Simple Harmonic Motion

Pendulum Equations

Period: T = 2π√(L/g) (small angle approximation)
Angular displacement: θ(t) = θ₀ cos(ωt + φ)
Angular frequency: ω = √(g/L)
For large angles, the period increases beyond the small-angle approximation

Spring-Mass Equations

Period: T = 2π√(m/k)
Displacement: x(t) = A cos(ωt + φ)
Angular frequency: ω = √(k/m)

Energy in SHM

Total energy: E = ½kA² (constant if undamped)
Kinetic energy: KE = ½mv²
Potential energy (spring): PE = ½kx²
Potential energy (pendulum): PE = mgL(1 - cosθ)
Energy oscillates between KE and PE while total E stays constant

Try This

Set a small angle and compare the period to T = 2π√(L/g)
Switch to Large Angle preset and observe the period increase
Add damping and watch the amplitude decay exponentially
Switch to Moon gravity and see the pendulum slow down

Pendulum & SHM Simulator

Controls

Values

The Physics of Simple Harmonic Motion

Pendulum Equations

Spring-Mass Equations

Energy in SHM

Try This

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