x(t) = A cos(ωt + φ) — Simulate, Explore, Practice & Quiz
Simple Harmonic Motion is a fundamental type of periodic oscillation where the restoring force acting on an object is directly proportional to its displacement from an equilibrium position and always directed towards that equilibrium. Described by the equation x(t) = A cos(ωt + φ), SHM appears throughout mechanical engineering, physics, and everyday life — from the oscillation of a spring-mass system to the swing of a pendulum. Understanding SHM is essential for studying vibrations, wave mechanics, alternating current circuits, and structural dynamics.
This interactive simulator lets you explore SHM with two classic systems: a vertical spring-mass and a simple pendulum. Adjust mass, spring constant, length, and amplitude to see how the motion parameters change in real time. The waveform chart displays displacement, velocity, acceleration, or energy graphs alongside the animated physical system.
For a spring-mass system, the angular frequency is ω = √(k/m), where k is the spring constant and m is the mass. The period is T = 2π/ω and the frequency is f = 1/T. Displacement, velocity, and acceleration are related by: v(t) = −Aω sin(ωt) and a(t) = −Aω² cos(ωt) = −ω²x(t). For a simple pendulum of length L, the angular frequency is ω = √(g/L) for small-angle oscillations.
In an ideal SHM system with no damping, total mechanical energy is conserved. At maximum displacement, all energy is potential (PE = ½kx²); at the equilibrium position, all energy is kinetic (KE = ½mv²). The total energy E = ½kA² remains constant throughout the cycle. The energy graph in this simulator shows the continuous exchange between kinetic and potential energy as the mass oscillates.
This SHM simulator is designed for mechanical engineering students, physics learners, technical trainees, and instructors teaching oscillatory motion. Whether you are preparing for exams, building intuition about periodic systems, or demonstrating concepts in a classroom, this tool provides an interactive, visual approach to mastering Simple Harmonic Motion.