Simple harmonic motion calculator

Simple harmonic motion is the undamped oscillation produced when a restoring force is linear in the displacement from equilibrium: a mass on a spring, a small-angle pendulum, a torsion oscillator, or the Cartesian projection of uniform circular motion. The position at time t is the closed-form sinusoid x(t) equals A times cosine of omega t plus phi, where A is the amplitude, omega is the angular frequency in radians per second, and phi is the phase constant in radians. This calculator evaluates that displacement for any finite amplitude, angular frequency, phase, and time, and also returns velocity and acceleration from the same argument.

What x(t) = A cos(ωt + φ) means

Simple harmonic motion is the unique undamped oscillation whose restoring force is strictly linear in the signed displacement from a single equilibrium. Under that Hooke-type law the Newtonian equation m times x double-dot plus k times x equals zero has the closed-form family x of t equals A cosine of omega t plus phi, with omega equal to the square root of k over m for a mass-spring pair.

The same sinusoid is the x-projection of a point that travels at constant speed around a circle of radius A, which is why textbooks introduce SHM through circular motion even when the laboratory system is a spring, a pendulum, or a torsion wire. Amplitude A is the maximum excursion from equilibrium and carries the length unit of the answer: metres in SI, or whatever consistent length unit you already used for A.

Angular frequency omega, in radians per second, is how fast the argument of the cosine advances; it is two pi times the ordinary frequency in hertz, and the period is two pi over the absolute value of omega when omega is not zero. Phase phi, in radians, sets the position at time zero: phi of zero starts at plus A, phi of pi starts at minus A, and phi of pi over two starts at zero heading negative.

Time t is the instant of evaluation in seconds; negative times are mathematically valid and simply run the oscillator backward. Damping, driving forces, and nonlinear springs are out of scope: they replace the single cosine with decaying envelopes, particular integrals, or numerical trajectories that belong on other landings.

Input contract, units, and error behaviour

Amplitude is the field the specification calls out: if amplitude is not a finite number the task fails and is not billed, rather than returning a hollow NaN displacement that a buggy client might treat as success. Infinity, NaN, and non-numeric strings are all rejected the same way. Negative amplitude is allowed because it is equivalent to adding pi to the phase; zero amplitude yields a rest displacement of zero at every time.

Zero angular frequency is also allowed and yields the constant displacement A cosine of phi, which is the static limit of an oscillator that is not oscillating. On success the body echoes the four cleaned inputs, the phase argument omega t plus phi, displacement x, velocity minus A omega sine of the argument, acceleration minus A omega squared cosine of the argument, and a formula string restating x of t equals A cosine of omega t plus phi.

Classic checks include A equals four, omega equals two, phi equals zero, t equals zero giving displacement four, and A equals two, omega equals one, phi equals pi, t equals zero giving displacement minus two.

Where an SHM displacement calculator earns its keep

Classrooms and tutoring platforms use a simple harmonic motion calculator so learners can check x of t for arbitrary A, omega, phi, and t without mixing sine and cosine conventions or forgetting that phase is in radians. Laboratory notebooks recording mass-spring traces often know amplitude from a ruler, period from a stopwatch (hence omega equals two pi over T), and a phase from the starting position; evaluating displacement at a chosen time is the first step before overlaying a theoretical curve on a photogate file.

Game and animation rigs that fake a bobbing buoy, a swaying sign, or a breathing HUD use the same cosine, and they need a pure function that never depends on wall-clock Date so a replay file stays bit-stable. Failed validation does not consume balance. Keep requests small, pin the rounded displacement in tests, and you get reproducible positions for homework, LMS plugins, and edge workers without standing up a numerical ODE solver of your own.