Simple Harmonic Motion Calculator
Simple Harmonic Motion (SHM)
Simple Harmonic Motion is a type of periodic motion where the restoring force is directly proportional to the displacement from equilibrium and always directed towards it. It is one of the most fundamental concepts in physics, describing oscillations of springs, pendulums, molecules, and even electrical circuits.
Key Characteristics
- Restoring Force: F = -kx (Hooke’s law for a spring).
- Displacement: x(t) = A cos(ωt + φ).
- Velocity: v(t) = -Aω sin(ωt + φ).
- Acceleration: a(t) = -Aω² cos(ωt + φ).
- Angular frequency: ω = √(k/m) for mass-spring systems.
- Time period: T = 2π√(m/k).
- Frequency: f = 1/T.
Examples
– Mass-spring system: A 2 kg mass attached to a spring (k = 200 N/m) has ω = √(200/2) = 10 rad/s and T = 2π/10 ≈ 0.63 s. – Pendulum: A simple pendulum of length L has T = 2π√(L/g). For L = 1 m, T ≈ 2 s.
Energy in SHM
The total energy in SHM is constant and given by:
E = (1/2) k A² = (1/2) m ω² A²
Energy oscillates between kinetic energy (KE) and potential energy (PE). At maximum displacement, PE is maximum; at equilibrium, KE is maximum.
Applications
- Spring-mass oscillations
- Pendulum clocks
- Vibrations in molecules
- Electrical LC circuits (analogous to SHM)
- Engineering vibration analysis
Historical Note
The study of oscillations dates back to Galileo, who observed pendulum motion in the 17th century. Hooke later formulated his law for springs, and Newton connected these principles to his laws of motion, creating the foundation for SHM analysis.