Simple Harmonic Motion Calculator

Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.

Key Concepts:

• Period (T): Time for one complete oscillation
• Frequency (f): Number of oscillations per unit time
• Angular Frequency (ω): Rate of change of phase angle
• Amplitude (A): Maximum displacement from equilibrium
• Phase Angle (φ): Initial phase of the oscillation

Key Formulas:

• T = 2π√(m/k) - Period (spring-mass)
• T = 2π√(L/g) - Period (pendulum)
• T = 2π√(I/κ) - Period (torsional)
• f = 1/T - Frequency
• ω = 2πf = √(k/m) - Angular frequency
• v_max = Aω - Maximum velocity
• a_max = Aω² - Maximum acceleration
• E = ½kA² - Total energy

SHM Equations:

• x(t) = A cos(ωt + φ) - Displacement
• v(t) = -Aω sin(ωt + φ) - Velocity
• a(t) = -Aω² cos(ωt + φ) - Acceleration

System Types:

• Spring-Mass: Mass attached to spring
• Simple Pendulum: Point mass on string
• Torsional Pendulum: Rotating mass on wire

Units:

• Period: seconds (s)
• Frequency: Hertz (Hz)
• Angular frequency: rad/s
• Amplitude: meters (m)
• Spring constant: N/m
• Energy: Joules (J)

Applications:

• Clocks and timekeeping
• Musical instruments
• Seismometers
• Atomic clocks
• Resonance systems