Simple Harmonic Motion Calculator

Simple Harmonic Motion Calculator. Enter your oscillating system's amplitude, frequency, and time to analyze its Simple Harmonic Motion. The calculator returns angular frequency, displacement, velocity, acceleration, and their maximum values — all based on the standard SHM equations. Also try the Simple Pendulum Calculator.

Results

Displacement at t

Angular Frequency (ω)

Period (T)

Velocity at t

Acceleration at t

Maximum Velocity (v_max)

Maximum Acceleration (a_max)

Results Table

What is Simple Harmonic Motion (SHM)?

Simple Harmonic Motion is a type of periodic oscillatory motion where the restoring force is directly proportional to the displacement from the equilibrium position and always directed toward it. Examples include a mass on a spring and a simple pendulum for small angles. The motion is described by sinusoidal functions of time. See also our Primary Result — Rotational Stiffness.

What equations does this calculator use?

The calculator uses the standard SHM equations: displacement y = A·sin(ωt + φ), velocity v = A·ω·cos(ωt + φ), and acceleration a = −A·ω²·sin(ωt + φ), where A is amplitude, ω = 2πf is angular frequency, t is time, and φ is the phase angle.

How do I calculate the angular frequency of an oscillating particle?

Angular frequency (ω) is calculated using ω = 2πf, where f is the frequency in Hz. It can also be expressed as ω = 2π/T, where T is the period. For a mass-spring system, ω = √(k/m) where k is the spring constant and m is the mass.

What is the phase angle and how does it affect the motion?

The phase angle (φ) represents the initial offset of the oscillation at time t = 0. A phase angle of 0° means the particle starts at the equilibrium position with maximum velocity, while 90° means it starts at maximum displacement with zero velocity. Changing the phase angle shifts the entire sine wave left or right in time.

What are the maximum velocity and acceleration in SHM?

The maximum velocity in SHM is v_max = A·ω, which occurs when the particle passes through the equilibrium position (displacement = 0). The maximum acceleration is a_max = A·ω², which occurs at the points of maximum displacement (the amplitude positions).

How is the period related to frequency in SHM?

Period (T) and frequency (f) are reciprocals of each other: T = 1/f. The period is the time taken to complete one full oscillation, measured in seconds. For example, a frequency of 2 Hz corresponds to a period of 0.5 seconds.

Can this calculator be used for spring-mass systems and pendulums?

Yes. For a spring-mass system, first compute the natural frequency using f = (1/2π)·√(k/m), then enter that frequency along with the amplitude and desired time. For a simple pendulum with small oscillations, use f = (1/2π)·√(g/L), where g is gravitational acceleration and L is the pendulum length.

What are the limitations of a Simple Harmonic Motion Calculator?

SHM calculators assume ideal conditions: no damping (no energy loss due to friction or air resistance), linear restoring force, and small oscillation amplitudes for pendulums. Real-world systems experience damping and nonlinear effects that cause the motion to deviate from pure SHM over time. You might also find our Frequency to Period Converter useful.