SHM displacement calculator

It applies the standard cosine form of the SHM equation and reports displacement in metres. Use it to check physics exercises, evaluate a vibration model, or generate a precise position for a simulation. Every input must be a finite number, so malformed values are rejected instead of producing a misleading result.

Enter quantities that describe the same oscillator

Amplitude sets the greatest distance from equilibrium and must be supplied in metres. Angular frequency describes how rapidly the phase advances and must be in radians per second, while time is entered in seconds. The phase constant is an angle in radians that fixes the oscillator's position within its cycle at time zero. Keep these units consistent: ordinary frequency in hertz is not the same quantity as angular frequency, although you can convert it with angular frequency equal to two times pi times frequency. A negative amplitude is mathematically accepted because the formula remains defined, but in most physical descriptions amplitude is a non-negative magnitude; a negative value is equivalent to shifting the phase by pi radians. Time may be negative when you want to extend the ideal model before its chosen origin. The calculator requires actual JSON numbers. Numeric-looking text, blank values, infinity, and non-numeric tokens are rejected so that an accidental formatting problem cannot silently change the physical interpretation.

Understand the cosine calculation

The calculator evaluates displacement as amplitude multiplied by the cosine of angular frequency times time plus phase. First, multiplying angular frequency by time produces the phase accumulated since the time origin. Adding the phase constant gives the oscillator's total phase at the requested instant. Taking the cosine maps that phase to a value between minus one and one, and multiplying by amplitude scales it to metres. At total phase zero, the oscillator is at positive maximum displacement. At pi over two it crosses equilibrium, at pi it reaches negative maximum displacement, and at two pi it returns to its starting point. This sign convention follows the cosine form of simple harmonic motion. A source that uses a sine form is not contradictory; sine and cosine descriptions differ by a phase shift. Results use JavaScript floating-point arithmetic, so values expected to be exactly zero may appear as extremely small residuals, such as a few quadrillionths of a metre, because pi and trigonometric values cannot generally be represented exactly in binary.

Use the result within the model's limits

The returned displacement is an instantaneous signed position relative to equilibrium, not the total distance travelled. A positive result lies on the side chosen as positive by your coordinate system, while a negative result lies on the opposite side. The calculation represents ideal simple harmonic motion: amplitude and angular frequency remain constant, and damping, external forcing, nonlinear stiffness, and measurement uncertainty are not included. That makes it well suited to textbook mass-spring systems, small-angle approximations, ideal vibration checks, and deterministic simulation inputs. For a real damped oscillator, the amplitude changes with time and this equation alone is incomplete. Likewise, velocity and acceleration require their own derivatives rather than treating displacement as a rate. When comparing a result with experimental data, confirm that the data use the same time origin and phase convention. Through the API, each calculation costs $0.002; the browser version can run the same deterministic formula locally. Because the operation uses no network, random source, or clock, identical numeric inputs always follow the same calculation path and produce the same serialized result.