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.
Displacement at t
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Angular Frequency (ω)
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Period (T)
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Velocity at t
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Acceleration at t
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Maximum Velocity (v_max)
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Maximum Acceleration (a_max)
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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.
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.
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.
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.
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).
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.
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.
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.