Simple Harmonic Motion Calculator
Computes the instantaneous displacement, velocity, or acceleration of an oscillating object undergoing simple harmonic motion. Ideal for physics students, engineers analysing springs, pendulums, or any periodic system.
About this calculator
Simple harmonic motion (SHM) describes any system where the restoring force is proportional to displacement. Given amplitude A, frequency f, time t, and phase angle φ (in radians), the three key quantities are: Displacement: x(t) = A · sin(2πft + φ). Velocity: v(t) = A · 2πf · cos(2πft + φ). Acceleration: a(t) = −A · (2πf)² · sin(2πft + φ). The term 2πf is the angular frequency ω (rad/s), so these can also be written x = A sin(ωt + φ), v = Aω cos(ωt + φ), and a = −Aω² sin(ωt + φ). Notice that velocity leads displacement by 90° and acceleration is exactly 180° out of phase with displacement. The negative sign in the acceleration formula reflects the restoring nature of the force: when displacement is positive, acceleration is negative, always pulling the object back toward equilibrium.
How to use
Suppose a mass on a spring has amplitude A = 0.05 m, frequency f = 2 Hz, phase angle φ = 0 rad, and you want the velocity at t = 0.1 s. Using the velocity formula: v = A · 2πf · cos(2πft + φ) = 0.05 × 2π × 2 × cos(2π × 2 × 0.1 + 0) = 0.1π × cos(0.4π) = 0.3142 × cos(72°) = 0.3142 × 0.3090 ≈ 0.0971 m/s. So at t = 0.1 s, the mass is moving at approximately 0.097 m/s. To find displacement at the same instant, use x = 0.05 × sin(0.4π) = 0.05 × 0.9511 ≈ 0.0476 m.
Frequently asked questions
What is the phase angle in simple harmonic motion and how does it affect the result?
The phase angle φ (phi) is a constant in radians that shifts the entire waveform along the time axis. A phase angle of 0 means the oscillation starts at x = 0 and moves in the positive direction. A phase angle of π/2 (≈1.5708 rad) means the object starts at maximum positive displacement. The phase angle is determined by the initial conditions of the system, such as where the object is and how fast it is moving at t = 0. Changing φ does not alter the amplitude, frequency, or energy of the motion.
How is simple harmonic motion different from damped harmonic motion?
In ideal simple harmonic motion, the amplitude remains constant forever because there is no energy loss. In damped harmonic motion, friction or resistance removes energy from the system, causing the amplitude to decrease over time, typically following an exponential decay envelope. Lightly damped systems (like a pendulum in air) oscillate many times before stopping, while critically damped systems return to equilibrium as fast as possible without oscillating at all. This calculator models ideal SHM only; it does not account for damping effects.
Why is the acceleration in simple harmonic motion always directed toward the equilibrium position?
The defining characteristic of SHM is that the net force (and therefore acceleration) is always proportional to displacement and directed opposite to it: F = −kx, which gives a = −ω²x. When the object is displaced to the right (positive x), the force pulls it left (negative a). This restoring behaviour is what creates the oscillation. The negative sign in the acceleration formula a(t) = −Aω²sin(ωt + φ) mathematically encodes this relationship, ensuring that acceleration and displacement are always 180° out of phase.