Simple Harmonic Motion
Calculate SHM displacement, velocity, and acceleration using y = A·sin(ωt). Free simple harmonic motion calculator with charts and breakdowns for physics students.
About This Calculator
The Simple Harmonic Motion (SHM) Calculator helps physics students, engineers, and enthusiasts compute the key parameters of an oscillating particle: displacement, velocity, acceleration, and angular frequency. Simply enter the amplitude, frequency, and time to get instant results with a detailed formula breakdown and interactive charts.
SHM describes periodic motion where the restoring force is proportional to displacement (Hooke's Law, F = -kx). The calculator uses the standard equations: displacement y = A·sin(ωt), velocity v = A·ω·cos(ωt), and acceleration a = -A·ω²·sin(ωt), where angular frequency ω = 2πf.
For a mass on an ideal spring, the angular frequency relates to spring constant k and mass m by ω = √(k/m). A simple pendulum approximates SHM for small angles (θ < 15°) with ω = √(g/L) where g is gravitational acceleration and L is pendulum length.
Key Concepts
Amplitude (A): Maximum displacement from equilibrium, measured in meters (m). Determines the energy of oscillation: E = ½kA² for a spring system.
Frequency (f): Number of complete oscillations per second, measured in hertz (Hz). Period T = 1/f.
Angular Frequency (ω): Rate of phase change in rad/s, ω = 2πf. This is the natural frequency parameter used in the SHM differential equation d²y/dt² + ω²y = 0.
Phase: The calculator assumes the particle starts at the equilibrium position moving upward at t = 0 (sine initial condition). For the cosine form (starting at maximum displacement), the displacement would shift from sin to cos.
Applications
SHM modeling is used across physics and engineering: mechanical vibrations, seismic analysis, acoustic waves, electronic oscillators (LC circuits), and quantum harmonic oscillators. Understanding these equations is fundamental for coursework in mechanics, waves, and differential equations.
Frequently Asked Questions
What is simple harmonic motion?
Simple harmonic motion (SHM) is a type of periodic oscillation where a particle moves back and forth along a straight line, with an acceleration that is always directed toward a fixed equilibrium point and proportional to its displacement from that point. Examples include a mass on a spring, a simple pendulum at small angles, and the vibration of a tuning fork.
What are the equations for simple harmonic motion?
The standard SHM equations are: displacement y = A·sin(ωt), velocity v = A·ω·cos(ωt), and acceleration a = -A·ω²·sin(ωt), where A is amplitude, ω = 2πf is angular frequency, f is frequency, and t is time. Angular frequency ω is measured in rad/s.
How is angular frequency calculated?
Angular frequency ω is calculated as ω = 2πf, where f is the ordinary frequency in hertz (Hz). It represents the rate of change of the phase of the sinusoidal waveform and is measured in radians per second (rad/s).
What is the difference between frequency and angular frequency?
Frequency f is the number of complete oscillations per second, measured in hertz (Hz). Angular frequency ω is the rate of change of angular displacement, measured in rad/s, and is related by ω = 2πf. Frequency tells you how many cycles per second, while angular frequency tells you how many radians per second the phase advances.
How does amplitude affect SHM?
Amplitude A is the maximum displacement from the equilibrium position. It determines the maximum velocity (v_max = A·ω) and maximum acceleration (a_max = A·ω²). Larger amplitude means larger energy but does not affect the frequency or period of oscillation for ideal SHM.
What are real-world examples of SHM?
Common examples of simple harmonic motion include a mass attached to an oscillating spring, a simple pendulum swinging at small angles, the vibration of guitar strings, the motion of a piston in an engine, seismic waves, and the oscillation of atoms in a crystal lattice. Many natural and engineered systems approximate SHM for small displacements.
How accurate are the results?
Results are computed using the standard SHM equations y = A·sin(ωt), v = A·ω·cos(ωt), and a = -A·ω²·sin(ωt) with values rounded to 4 decimal places. The calculator uses JavaScript double-precision floating-point arithmetic, giving accuracy to approximately 15 significant digits.
Can I share my SHM calculation results?
Yes, the URL saves your input values automatically so you can bookmark or share the exact calculation. Any calculator with the same parameters will reproduce identical results.