Simple Harmonic Motion & Pendulum Calculator
Calculate simple pendulum periods (T = 2π√(L/g), Borda’s finite-amplitude correction, gravity inversion), spring-mass oscillators (series & parallel effective stiffness, vertical equilibrium stretch), and general harmonic waveforms (displacement, velocity, acceleration phase shifts & conserved mechanical energy) with interactive simulations.
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Simple Pendulum Inputs
T = 2π√(L/g)At 5°, the finite-amplitude period differs by about 0.048%. Forward correction includes θ⁴; gravity inversion retains θ².
Pendulum Motion Simulator
θ(t): 0.00° / v: 0.00 m/sKey Results & Physical Quantities
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Simple Harmonic Motion & Pendulum Calculator
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How to use
- 1
Choose oscillator model
Select Simple Pendulum (length & gravity), Spring-Mass Oscillator (series/parallel springs), or SHM Waveforms & Energy.
- 2
Enter parameters or targets
Set pendulum length L, gravity g (celestial presets available), initial swing angle θ₀, spring constant k, or mass m. Invert g from measured periods.
- 3
Inspect period, motion & energy
Review period T (small-angle & Borda-corrected), frequency f, maximum velocity v_max, tension/acceleration, energy breakdown, and live animation.
Features
- High-precision calculation of simple pendulum period T₀ = 2π√(L/g), frequency, and angular frequency
- Borda finite-amplitude correction T_corr and percentage deviation for large swing angles
- Inversion of local gravitational acceleration g = 4π²L / T² from measured lab period
- Calculation of required pendulum length L for clockmaking and target periods (such as seconds pendulums)
- Gravitational acceleration presets for Earth (standard, equator, poles, Tokyo), Moon, Mars, and Jupiter
- Spring-mass oscillator effective stiffness and period for single, parallel, and series springs
- Equilibrium displacement x₀ = mg/k calculation for vertical spring-mass systems
- Simultaneous computation of maximum velocity v_max = Aω, acceleration a_max, and cord tension F_max
- Real-time multi-waveform graph displaying displacement x(t), velocity v(t), and acceleration a(t)
- Conserved mechanical energy meter displaying real-time kinetic K and potential U ratios
- Realistic presets including Foucault pendulum, seconds pendulum, lunar gravity, and vehicle suspension
- One-click report text copy tailored for laboratory reports and academic submissions
Use cases
High school & college physics coursework
Eliminate confusion between pendulum and spring formulas, verifying phase shifts, velocities, and accelerations.
Physics lab reports on gravity determination
Invert local gravity g from experimental pendulum periods, comparing small-angle vs Borda-corrected values.
Mechanical vibration & suspension engineering
Design and compare series vs parallel spring constants and natural resonant frequencies for mechanical systems.
Details
Simple Harmonic Motion (SHM) is the fundamental periodic motion executed by an object subject to a linear restoring force F = -kx. It underpins mechanics, acoustics, vibration engineering, and quantum physics.
A simple pendulum consists of a point bob of mass m suspended from a pivot by a light string of length L. For small angular displacements (θ ≪ 1 rad, typically ≤ 5°), the restoring force mg sin θ ≈ mgθ yields the classic small-angle period T₀ = 2π√(L/g), exhibiting isochronism (independence of mass and amplitude). At larger swings, non-linear effects lengthen the period according to complete elliptic integrals, accurately approximated by Borda’s series T ≈ T₀(1 + θ₀²/16). The pendulum is also widely used in physics labs to determine local gravitational acceleration via g = 4π²L / T².
A spring-mass oscillator has a period T = 2π√(m/k). When combining two springs, parallel arrangement increases the stiffness to k = k₁ + k₂ (shortening the period), whereas series arrangement decreases stiffness via 1/k = 1/k₁ + 1/k₂ (lengthening the period). When suspended vertically, gravity displaces the equilibrium center by x₀ = mg/k below natural length without affecting the angular frequency ω = √(k/m).
The general solution x(t) = A cos(ωt + φ) leads to velocity v(t) = -Aω sin(ωt + φ) which leads displacement by 90° (π/2 radians), while acceleration a(t) = -Aω² cos(ωt + φ) = -ω²x is in exact opposite phase (180°). Kinetic energy K = ½mv² and potential energy U = ½kx² continuously convert into one another, while total mechanical energy E = K + U = ½kA² remains rigorously conserved.
This tool integrates all exact equations, Borda corrections, and parameter inversions with smooth interactive SVG animations, phase waveform plots, energy conservation meters, and one-click report export.
FAQ
Does the mass of the bob affect the period of a simple pendulum?
No. In the small-angle regime, the period T = 2π√(L/g) is completely independent of the bob’s mass m. A heavy bob and a light bob of the same length L oscillate with identical periods. However, tension in the cord and mechanical energy do scale with mass.
What is the small-angle approximation and why does the period change at large angles?
The restoring force is strictly -mg sin θ. When θ is small in radians (typically ≤ 5°), sin θ ≈ θ, yielding true simple harmonic motion. For larger angles, sin θ < θ, meaning the restoring force is weaker than linear, which causes the pendulum to swing more slowly and lengthens the period. At 30°, the period is ~1.7% longer than predicted by 2π√(L/g).
Why do laboratory pendulum experiments time 20 to 50 oscillations instead of just one?
Human stopwatch reaction time introduces ~0.1 to 0.2 s of measurement error. Timing a single 2-second swing yields a 5-10% error. Timing 20 oscillations (~40 s) and dividing by 20 reduces the relative timing uncertainty twentyfold to ~0.25-0.5%, yielding a much more accurate gravity estimate.
If a spring is cut in half, what happens to its spring constant?
Its spring constant doubles (it becomes twice as stiff). A full spring of length L can be viewed as two half-length springs connected in series: 1/k = 1/(2k) + 1/(2k). Halving the length halves the stretch under a given force.
What does it mean that velocity leads displacement by 90° in SHM?
At maximum displacement (x = +A), the object momentarily stops (v = 0). As it passes through the equilibrium center (x = 0), its speed reaches maximum (|v| = Aω). In wave plots, the velocity peak precedes the displacement peak by one-quarter of a cycle (T/4, or 90° / π/2 radians), meaning velocity leads displacement.
Related guide
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Verified: Verified pendulum period (L=1m, g=9.807 -> T=2.006s), Borda correction (+1.71% at 30°), gravity inversion (L=1m, T=2.006s -> g=9.807), spring oscillator (m=250g, k=25N/m -> T=0.628s, vmax=0.5m/s), series/parallel effective stiffness, vertical stretch, and conserved mechanical energy with automated tests
Did you know?
Galileo Galilei is said to have noticed the isochronism of the pendulum (period is independent of amplitude) by timing the swings of a chandelier in Pisa Cathedral against his pulse. This isochronism is an approximation valid for small angles; at larger swings, the period lengthens slightly due to finite-amplitude effects.