Simple Harmonic Motion Calculator
Calculate displacement, velocity, and acceleration for oscillating particles
Calculate Simple Harmonic Motion
Maximum displacement from equilibrium position
Number of oscillations per unit time
Time at which to calculate motion parameters
Initial phase offset (0 for standard SHM)
Motion Results at t = 0.000 s
Equations used:
• Displacement: y = A × sin(ωt + φ)
• Velocity: v = Aω × cos(ωt + φ)
• Acceleration: a = -Aω² × sin(ωt + φ)
• Angular frequency: ω = 2πf
Motion Analysis
Example Calculation
Spring-Mass System Example
Given: A = 15 mm, f = 1 Hz, t = 1.4 s
Find: Displacement, velocity, and acceleration
Solution
1. ω = 2πf = 2π(1) = 6.28 rad/s
2. y = 0.015 × sin(6.28 × 1.4) = 8.82 mm
3. v = 0.015 × 6.28 × cos(6.28 × 1.4) = -76.25 mm/s
4. a = -0.015 × (6.28)² × sin(6.28 × 1.4) = -348 mm/s²
SHM Characteristics
Common SHM Examples
Simple pendulum (small angles)
Mass-spring system
Vibrating molecules
Sound waves and musical instruments
LC oscillating circuits
Understanding Simple Harmonic Motion
What is Simple Harmonic Motion?
Simple Harmonic Motion (SHM) is a type of periodic motion where a particle oscillates back and forth about an equilibrium position. The restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction.
Key Properties
- •Motion is periodic and sinusoidal
- •Acceleration is proportional to displacement
- •Total mechanical energy is conserved
- •Maximum velocity occurs at equilibrium
SHM Equations
y(t) = A sin(ωt + φ)
v(t) = Aω cos(ωt + φ)
a(t) = -Aω² sin(ωt + φ)
ω = 2πf = 2π/T
- A: Amplitude (maximum displacement)
- ω: Angular frequency (rad/s)
- f: Frequency (Hz)
- T: Period (s)
- φ: Phase constant (rad)
Energy in SHM: The total energy E = ½kA² remains constant, continuously converting between kinetic and potential energy.