Simple Harmonic Motion Calculator

Simple harmonic motion (SHM) describes back-and-forth oscillation where the restoring force is directly proportional to displacement. The core kinematic equations governed by this calculator include: displacement x(t) = A * sin(omega * t), where A is amplitude and omega is angular frequency; velocity v(t) = A * omega * cos(omega * t); and acceleration a(t) = -A * omega^2 * sin(omega * t). Additionally, angular frequency relates directly to standard frequency via omega = 2 * pi * f.

Consider a mass oscillating on a frictionless spring with an amplitude (A) of 0.5 meters and an angular frequency (omega) of 4 radians per second. We want to find the displacement, velocity, and acceleration at time (t) = 0.5 seconds. First, compute displacement: x = 0.5 * sin(4 * 0.5) = 0.5 * sin(2). Since sin(2 radians) is approximately 0.9093, displacement equals 0.4546 meters. Next, calculate velocity: v = 0.5 * 4 * cos(2) = 2 * (-0.4161) = -0.8322 meters per second. Finally, determine acceleration: a = -(4^2) * 0.4546 = -16 * 0.4546 = -7.2736 meters per second squared.

Always ensure your trigonometric mode matches your input data; angular frequency and time inputs typically require radians rather than degrees. Pay close attention to negative signs in acceleration formulas, as they signify that the restoring force always acts in the direction opposite to the displacement. When analyzing physical experiments, double-check your units to maintain consistency between meters, seconds, and radians.