Simple Harmonic Motion Calculator
Analyze oscillating systems like springs and pendulums. Find the period, frequency, or displacement at any time with this simple calculator.
ω = √(k/m) | x(t) = A × cos(ωt) | T = 2π√(m/k)
About Simple Harmonic Motion:
SHM occurs when a restoring force is proportional to displacement. Common examples include mass-spring systems and pendulums (for small angles). The motion is sinusoidal with constant period independent of amplitude.
How the Simple Harmonic Motion Calculator Works
Select your oscillating system: mass-spring or simple pendulum. For mass-spring, enter the mass and spring constant. For pendulum, enter the length and (optionally) the release angle.
The calculator computes the period T, frequency f, and angular frequency ω. For mass-spring: T = 2π√(m/k). For pendulum (small angles): T = 2π√(L/g). You can also calculate displacement, velocity, and acceleration at any time.
Results show all oscillation parameters with units. Graphs display position, velocity, and acceleration versus time. The phase relationships are visible: velocity leads position by 90°, acceleration is 180° out of phase with position.
When You'd Actually Use This
Clock and timekeeping design
Design pendulum clocks. Pendulum length determines period. A 1-meter pendulum has ~2 second period, perfect for clock mechanisms.
Vehicle suspension analysis
Analyze car suspension oscillations. Spring constant and vehicle mass determine natural frequency. Avoid resonance with road inputs.
Seismometer calibration
Understand earthquake detection. Seismometers use mass-spring systems tuned to detect ground motion at specific frequencies.
Building vibration analysis
Calculate natural frequency of structures. Buildings oscillate during earthquakes. Engineers design to avoid resonance with seismic frequencies.
Musical instrument physics
Analyze vibrating strings and air columns. String instruments follow similar harmonic motion principles. Frequency determines pitch.
Physics education labs
Verify SHM equations experimentally. Measure period for different masses or lengths. Compare experimental results to theoretical predictions.
What to Know Before Using
Period is independent of amplitude (for small oscillations).Galileo discovered this for pendulums. A pendulum's period doesn't depend on swing size (for small angles). This isochronism enables accurate clocks.
Pendulum formula assumes small angles.T = 2π√(L/g) is accurate for angles < 15°. Larger angles require correction. At 30°, period is ~1.7% longer than the formula predicts.
Mass doesn't affect pendulum period.Heavier pendulums swing at the same rate as light ones (same length). Gravity accelerates all masses equally.
Real systems have damping.Friction and air resistance cause amplitude to decay. The period is slightly affected. Pure SHM assumes no damping.
Pro tip: For large pendulum amplitudes, use the correction: T ≈ 2π√(L/g) × [1 + θ₀²/16], where θ₀ is in radians. At 30° (0.52 rad), this adds about 1.7% to the period.
Common Questions
What's the difference between period and frequency?
Period (T) is time per cycle (seconds). Frequency (f) is cycles per second (Hz). They're reciprocals: f = 1/T.
Why doesn't mass affect pendulum period?
Gravity provides restoring force proportional to mass (F = mg sin θ). In F = ma, mass cancels. All objects fall at the same rate.
What's angular frequency?
ω = 2πf = 2π/T. Measured in radians/second. One complete cycle is 2π radians. Useful in equations: x(t) = A cos(ωt + φ).
How long should a 1-second pendulum be?
For T = 2 s (1 s each way), L = gT²/(4π²) ≈ 1 meter. Grandfather clocks use ~1 m pendulums for this reason.
What happens at resonance?
Driving at the natural frequency causes amplitude to grow dramatically. This can be useful (pushing swings) or destructive (bridge collapse).
Does gravity affect spring oscillations?
Gravity shifts the equilibrium position but doesn't change the period. A vertical spring oscillates at the same frequency as a horizontal one.
What's the energy in SHM?
Total energy is constant: E = ½kA². Energy oscillates between kinetic (maximum at equilibrium) and potential (maximum at extremes).
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