Sit on a playground swing and give one push. You glide forward, slow,
pause, and come back — over and over. Here's the surprising part: whether you
swing in a big arc or a small one, each round trip takes almost the same
amount of time. A heavier friend on the same swing keeps the same rhythm
too. The only thing that really changes the timing is the length
of the ropes: a long swing is slow and lazy, a short one is quick and busy.
That steady back-and-forth beat is what a pendulum is all about.
Build it up: what sets the timing?
One full back-and-forth is called the period, $T$. Two
things control it: the length $L$ of the string and the strength of gravity
$g$ pulling the bob down. Longer string → longer period; stronger gravity →
shorter period. For gentle swings these combine into one clean formula:
Period of a simple pendulum (small swing)
$$T=2\pi\sqrt{\frac{L}{g}}$$
Try a real number: a $1\text{ m}$ string on Earth ($g=9.81\text{ m/s}^2$)
gives $T=2\pi\sqrt{1/9.81}\approx2.0\text{ s}$ — about one second to swing out
and one to swing back. Notice what's missing: the mass $m$ and the
size of the swing don't appear at all.
Go deeper (AP / intro-college): why, and where it breaks
Gravity pulls straight down, but only the part along the arc,
$-mg\sin\theta$, pushes the bob back toward the bottom. Newton's law along the
arc gives $\ddot{\theta}=-\frac{g}{L}\sin\theta$. For small angles
$\sin\theta\approx\theta$, so the restoring pull is proportional to the
displacement — the signature of simple harmonic motion — and
out pops $T=2\pi\sqrt{L/g}$. Because $m$ cancels (gravity pulls harder on a
heavy bob but inertia resists exactly as much), mass drops out. At large
angles $\sin\theta<\theta$, so the restoring pull is a little weaker
than SHM assumes and every swing takes longer: about $1.7\%$ longer at
$30^\circ$ and $18\%$ longer at $90^\circ$. The sliders map straight onto the
equation — $L$, $g$, the starting angle $\theta_0$, and the damping $b$ that
drains energy each cycle.
Try this in the simulation above
Start with a small $\theta_0$ and watch the Δ% error badge
read near zero, then drag $\theta_0$ up toward $170^\circ$ and see it climb as
the small-angle formula breaks down. Switch the planet preset to
🌕 Moon ($g=1.62$) and watch the period nearly double for the
same length. Finally set Damping $b=0$ and the pendulum swings
forever; nudge $b$ high and it barely makes it back to center.
Tangential restoring force. Of the bob's weight $mg$, only the component along the arc, $F_t = -mg\sin\theta$, acts as the restoring force. String tension balances the radial component.
String tension. Radial balance: $T_s=m(g\cos\theta+L\dot\theta^2)$. Maximum tension occurs at $\theta=0$ (bottom of swing).
4
Linear damping. Air resistance adds $-\frac{b}{m}\dot\theta$. Damped natural frequency: $\omega_d=\sqrt{\omega_0^2-(b/2m)^2}$, giving a slightly longer period.
5
Small-angle SHO. For $\theta_0\lesssim15°$, $\sin\theta\approx\theta$. Solution: $\theta(t)=\theta_0 e^{-bt/2m}\cos(\omega_d t+\phi)$. Period $T=2\pi\sqrt{L/g}$ — independent of mass and amplitude.
6
Large-angle correction. Exact period $T=2\pi\sqrt{L/g}\!\left(1+\tfrac{1}{16}\theta_0^2+\tfrac{11}{3072}\theta_0^4+\cdots\right)$ — always larger than SHO formula.
7
Numerical method. Simulation uses 4th-order Runge–Kutta (RK4) at $\Delta t=0.003\,\text{s}$, accurate for all amplitudes and all modes including the chaotic double pendulum.
Ref: Halliday, Resnick & Walker — Fundamentals of Physics, 10th Ed., §15-4 & §15-6; Serway & Jewett — Physics for Scientists and Engineers, 8th Ed., §15.5; Kleppner & Kolenkow — An Introduction to Mechanics, 2nd Ed., §7.4.
Section 04
Frequently Asked Questions
A point mass (bob) on a massless inextensible string, pivoted at a fixed point. The full nonlinear ODE is solved with RK4 — accurate at any amplitude. Five modes: single, comparison, chaotic double, driven/resonance, and quiz. Drag the bob with mouse or touch to set initial angle interactively.
Mass cancels when dividing Newton's law by $mL$: both gravitational force and inertia scale with $m$, so the ratio (which governs dynamics) is independent of $m$. This mirrors Galileo's free-fall result. Mass matters only when damping is present, via the $b/m$ term — lighter bobs lose energy faster.
No — valid only for small amplitudes. The exact period (an elliptic integral) is always larger: ~4% longer at 45°, ~18% at 90°. The simulation displays the real-time % deviation between the small-angle formula and the measured period — increase θ₀ to see it grow.
Grandfather clocks exploit isochronism (Huygens, 1656). Foucault's Pendulum in Paris proved Earth's rotation (1851). Taipei 101's 660-tonne tuned mass damper is a giant pendulum absorbing wind energy. Seismometers, ballistic pendulums, and even human walking gait are governed by the same physics.
A plot of $\omega$ vs $\theta$. For undamped motion it is a closed ellipse (repeating orbit). Damping spirals it inward toward the origin. Driven oscillations can produce strange attractors. The double pendulum's portrait is a chaotic tangle — capturing the system's qualitative behaviour at a glance.
The coupled nonlinear equations have no closed-form solution for large amplitudes. Tiny differences in initial conditions grow exponentially — deterministic chaos. No randomness is involved; the future is determined by Newton's laws, but extreme sensitivity makes long-term prediction impossible in practice.
At the endpoints, KE=0 and PE is maximum ($mgh$, $h=L(1-\cos\theta)$). At the bottom, all PE converts to KE, giving maximum speed $v_\max=\sqrt{2gL(1-\cos\theta_0)}$. With damping, total energy decreases each cycle — watch the Energy graph amplitudes shrink together.
Resources: Khan Academy — Pendulums (khanacademy.org); HyperPhysics — Simple Pendulum (hyperphysics.phy-astr.gsu.edu); MIT OCW 8.03 — Oscillations lectures.
Section 05
Common Misconceptions
❌ A heavier bob swings faster — gravity pulls it harder.
✅ Mass cancels in the equation of motion. Greater gravitational force is exactly offset by greater inertia — $T$ is independent of $m$. Mass only affects the period via the $b/m$ damping term when damping is present.
📖 Halliday, Resnick & Walker, 10th Ed., §15-4; Arons — A Guide to Introductory Physics Teaching, Ch. 7.
❌ $T=2\pi\sqrt{L/g}$ is exact for any starting angle.
✅ Valid only for small amplitudes. At 30° the formula underestimates the period by ~2%; at 90° by ~18%. A clock that swings at 30° but is timed by the small-angle formula runs slow by roughly 25 minutes a day (~1500 s) — which is exactly why real pendulum clocks keep the swing to just a few degrees. The simulation's error badge quantifies this in real time.
❌ The restoring force points straight down — it is the full weight $mg$.
✅ Only the tangential component $-mg\sin\theta$ acts as the restoring force, directed along the arc toward equilibrium. The radial component is balanced by string tension. Confusing these leads to incorrect free-body diagrams.
❌ The bob moves fastest at the endpoints and slowest at the bottom.
✅ The opposite. At the endpoints KE=0 (momentarily at rest). At the bottom all PE has become KE, giving $v_\max=\sqrt{2gL(1-\cos\theta_0)}$. Watch the KE graph peak every time the bob passes vertical.
📖 Serway & Jewett, 8th Ed., §15.5, Example 15.5.
❌ Damping only reduces amplitude — the period is unchanged.
✅ Damping also slightly lengthens the period: $\omega_d=\sqrt{\omega_0^2-(b/2m)^2}<\omega_0$. Heavy damping can prevent oscillation entirely (overdamped). Set $b$ high in the simulation and observe the pendulum creep back without swinging.
❌ The double pendulum is unpredictable because the equations are unknown.
✅ The equations are known exactly from Newton's laws. It is chaotic — not random — due to extreme sensitivity to initial conditions. Perfect knowledge of the initial state would give a perfectly determined future; in practice, any tiny uncertainty grows exponentially.
📖 Strogatz — Nonlinear Dynamics and Chaos, Ch. 9; Taylor — Classical Mechanics, Ch. 11.
Misconception research: Arnold Arons — A Guide to Introductory Physics Teaching, Wiley, 1990; Halloun & Hestenes (1985), Am. J. Phys. 53, 1056; Phys. Rev. Physics Education Research — pendulum concept inventory studies.