Large-Amplitude Pendulum Simulator — Period vs Amplitude
Compute the small-angle period T0 = 2*pi*sqrt(L/g) and the exact period T = 4*sqrt(L/g) * K(sin(theta0/2)) via the complete elliptic integral of the first kind, in real time, from pendulum length L, amplitude theta0, gravity g and mass m. The tool also reports the period error and the gravitational potential energy m*g*L*(1 - cos(theta0)), with a live pendulum animation and a period-error-vs-amplitude curve that makes "where isochronism breaks down" intuitive at a glance.
Parameters
Pendulum length L
m
Amplitude theta0
deg
Gravity g
m/s^2
Mass m
kg
Defaults: L=1.0 m, theta0=45 deg, g=9.81 m/s^2, m=1.0 kg. The pendulum animation uses the exact period T to display one oscillation per period in real time, and dashed rods mark the maximum amplitude positions (+/- theta0). Mass m does not affect the period but does affect the gravitational potential energy.
Results
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Small-angle period T0
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Exact period T
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Period error (T-T0)/T0
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Potential energy E
Pendulum animation
Pendulum of length L and mass m hung from a fixed pivot. The blue bob shows the current angle, and the dashed orange rods mark the maximum-amplitude stopping positions +/- theta0. When playing, the bob oscillates in real time with period T.
Period error vs amplitude theta0
x: amplitude theta0 (deg) [0–175]; y: period error (%). Blue curve = exact result from the elliptic integral; dashed orange = first series term theta0^2/16. Yellow marker shows the current operating point (theta0, error). The curve diverges as theta0 approaches 180 deg.
Theory & Key Formulas
The pendulum period takes a very different form in the small-amplitude and exact regimes. Both expressions are shown side by side.
Small-angle period (isochronism):
$$T_{0} = 2\pi\sqrt{\dfrac{L}{g}}$$
Exact period (complete elliptic integral of the first kind):
$$T \approx T_{0}\left(1+\dfrac{\theta_{0}^{2}}{16}+\dfrac{11\theta_{0}^{4}}{3072}+\dfrac{173\theta_{0}^{6}}{737280}+\cdots\right),\qquad E = m g L (1-\cos\theta_{0})$$
$L$ is the pendulum length [m], $\theta_{0}$ is the amplitude [rad], $g$ is the gravitational acceleration [m/s^2] and $m$ is the mass [kg]. $T_{0}$ is independent of mass and amplitude (isochronism), while $T$ grows with $\theta_{0}$ and diverges as $\theta_{0}\to\pi$. The potential energy $E$ is the value at the maximum height (at amplitude $\theta_{0}$).
What the Large-Amplitude Pendulum Simulator does
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With the defaults at theta0=45 deg the tool shows T0=2.006 s, T=2.086 s and 4.00% error. Just 45 deg gives a 4% discrepancy? I was taught the pendulum is "isochronous".
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Great catch. Isochronism only holds in the small-angle regime where sin(theta) is well approximated by theta — roughly theta0 below 10 deg. At theta0=45 deg sin(theta) is smaller than theta, so the restoring torque is weaker and the period stretches. The leading series correction theta0^2/16 with theta0=0.785 rad already gives 3.85%, very close to the exact 4.00% the tool reports via the AGM-based elliptic integral. Galileo's 1602 observation of "isochronism" was made on amplitudes of only a few degrees, so it was always an approximate law, not a universal one.
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If I push theta0 up to 175 deg the error explodes to over 1000% and the bob nearly stops at the top of the screen. What's happening physically?
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As theta0 approaches 180 deg, K(1) diverges and the period itself becomes infinite. Physically that means it takes infinite time to reach the upright unstable equilibrium — energy conservation forces the kinetic energy to zero exactly when theta=180 deg, and any small perturbation grows exponentially. In phase space this trajectory is the "separatrix" that separates oscillating orbits from rotational ones. It's a classic example of "slowing down" near a saddle point.
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The blue (exact) and dashed orange (1st series term) curves diverge sharply from about theta0=60 deg. How many terms do I need for practical accuracy?
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The series T = T0*(1 + theta0^2/16 + 11*theta0^4/3072 + 173*theta0^6/737280 + 22931*theta0^8/1321205760 + ...) has radius of convergence theta0=pi. At theta0=45 deg two terms reach 0.005% accuracy and four terms reach machine precision. At theta0=120 deg (2.09 rad) the first term gives 27.4% versus the exact 31.7%, and you need 5–6 terms. In practice, two or three terms are fine below 60 deg, and the AGM-based elliptic integral wins everywhere else. This tool uses AGM globally and reaches machine precision in about 30 iterations.
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Changing the mass m only updates E, not T0 or T. Why doesn't the pendulum period depend on mass?
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From the equation of motion m*L*theta'' = -m*g*sin(theta), the mass m cancels: theta'' = -(g/L)*sin(theta). That cancellation is the same statement that "inertial mass equals gravitational mass" (the equivalence principle) — exactly the property Galileo demonstrated with falling balls and that became the starting point of Einstein's general relativity. The potential energy E = m*g*L*(1 - cos(theta0)) is proportional to m, so for gravity-based energy storage the mass matters a lot. Switzerland-based Energy Vault commercialises MWh-scale gravity storage by raising and lowering 35-tonne concrete blocks.
Physical model and key equations
An ideal pendulum of length $L$ and mass $m$ swinging in gravity $g$ obeys Newton's second law: $m L \ddot{\theta} = -m g \sin\theta$, that is, $\ddot{\theta} = -(g/L)\sin\theta$. The small-angle approximation $\sin\theta\approx\theta$ gives the harmonic-oscillator equation $\ddot{\theta}+(g/L)\theta=0$, the natural angular frequency $\omega_{0}=\sqrt{g/L}$ and the period $T_{0}=2\pi/\omega_{0}=2\pi\sqrt{L/g}$, independent of mass and amplitude.
For finite amplitudes we keep $\sin\theta$ exact. Energy conservation $\tfrac{1}{2}L^{2}\dot{\theta}^{2}+gL(1-\cos\theta)=gL(1-\cos\theta_{0})$ yields $\dot{\theta}=\pm\sqrt{(2g/L)(\cos\theta-\cos\theta_{0})}$. Integrating from 0 to $\theta_{0}$ and multiplying by 4 gives the period:
where $K(k)=\int_{0}^{\pi/2}d\varphi/\sqrt{1-k^{2}\sin^{2}\varphi}$ is the complete elliptic integral of the first kind. As $k=\sin(\theta_{0}/2)\to 1$ we have $K(k)\to\infty$, so the period diverges. The series expansion is $T=T_{0}(1+\theta_{0}^{2}/16+11\theta_{0}^{4}/3072+173\theta_{0}^{6}/737280+\cdots)$, whose leading correction $\theta_{0}^{2}/16$ is the canonical "large-amplitude correction". The potential energy at the highest point is $E=mgL(1-\cos\theta_{0})$; it scales with mass but does not appear in the period. The tool evaluates $K(k)$ to machine precision using the arithmetic-geometric mean (AGM) method.
Real-world applications
Precision pendulum clocks: Since Huygens (1656) pendulum clocks have used escapements to keep the swing amplitude almost constant, exploiting isochronism. Riefler and Shortt astronomical clocks held the amplitude near 1.5–2 deg, suppressing the period error to (0.026)^2/16 ~ 4.3e-5 ~ 0.0043% and delivering 1/100 s/day accuracy — they were the world's most precise time standard for ~300 years until atomic clocks. Enter theta0=2 deg in the tool to see this 0.0076% error reproduced.
Earthquake response and tuned mass dampers (TMDs): The first mode of a tall building behaves like a pendulum, and pendulum TMDs on the roof oscillate in counter-phase to absorb the structural motion. Taipei 101 (660 t pendulum) and the Shanghai Tower (1000 t) are well-known examples. Designers must account for the 4%-class period drift of the TMD pendulum under design-level seismic amplitudes — the tool's large-amplitude correction maps directly onto this problem, and full CAE work uses nonlinear time integration (Newmark-beta, central differences) keeping sin(theta) exact.
Circadian and biological clocks: Mammalian circadian rhythms are nonlinear oscillators with amplitude-dependent periods — light stimuli that shift the amplitude can shift the period by 0.5–1 hour, a hallmark "breakdown of isochronism" that shares the same mathematical structure as the large-amplitude pendulum. Hodgkin-Huxley neuron firing models follow the same template and inform the theoretical analysis of epileptic seizures.
Gravity measurement and the shape of the Earth: The Kater reversible pendulum (1818) determines $g$ from $g=4\pi^{2}L/T^{2}$ with 0.001% precision and was the workhorse of geodetic gravity surveys well into the 20th century. With $L=1$ m and $T=2.006$ s the tool recovers $g=4\pi^{2}\cdot 1/2.006^{2}=9.81$ m/s^2. Modern gravity is measured with superconducting gravimeters and free-fall absolute gravimeters, but the pendulum idea lives on in cavity detection, ore-body surveys and crustal-strain monitoring.
Common misconceptions and caveats
The most common misconception is that "the pendulum period is always T = 2*pi*sqrt(L/g) regardless of amplitude". This is only the small-angle limit. The exact answer grows to 0.19% at theta0=10 deg, 4.00% at 45 deg, 18.1% at 90 deg, 32% at 120 deg and 290% at 170 deg. The tool's error curve makes "isochronism breaks down beyond ~10 deg" obvious. Textbooks always state "small oscillations" as a hypothesis; check the amplitude scale before applying the simple formula.
The next most common mistake is to assume "making the mass heavier (or lighter) changes the period". For an ideal pendulum the mass cancels out of the equation of motion, so $T$ does not depend on $m$ (the equivalence principle in action). The tool confirms this: moving the m slider only updates $E$ while $T_{0}$ and $T$ stay fixed. Real pendulums can break this law via air drag (it scales with surface but the inertia scales with mass, so light bobs damp faster) or non-negligible cord mass (a physical-pendulum model is then needed).
Finally, do not assume "the elliptic integral can only be computed by approximation". There is no closed elementary form, but the arithmetic-geometric mean method converges quadratically — the number of correct digits doubles per iteration — and reaches machine precision in roughly 30 iterations. Python's scipy.special.ellipk and Boost's boost::math::ellint_1 use AGM internally. This tool runs up to 50 iterations and remains stable to theta0=174.999 deg. The series expansion is fine below 60 deg but needs many terms for large amplitudes, so AGM is the practical workhorse.
Frequently asked questions
The equation of motion d^2(theta)/dt^2 = -(g/L) * sin(theta) contains the nonlinear term sin(theta). For small amplitudes sin(theta) is well approximated by theta, giving the amplitude-independent period T0 = 2*pi*sqrt(L/g) (isochronism). For large amplitudes sin(theta) < theta, so the restoring torque is weaker and the period grows. The exact relation is T = 4*sqrt(L/g) * K(sin(theta0/2)) with the complete elliptic integral of the first kind. With the defaults L=1.0 m, theta0=45 deg, g=9.81 m/s^2 the tool returns T0=2.006 s, T=2.086 s and a 4.00% error.
K(k) has no closed-form elementary expression, but the arithmetic-geometric mean (AGM) method reaches machine precision in just a few iterations. Starting from a0=1 and b0=sqrt(1-k^2), repeat a_{n+1}=(a_n+b_n)/2 and b_{n+1}=sqrt(a_n*b_n) to obtain the limit M(1, sqrt(1-k^2)); then K = pi/(2*M). This tool uses up to 50 AGM iterations and remains accurate up to theta0=175 deg. The series T = T0*(1 + theta0^2/16 + 11*theta0^4/3072 + ...) converges slowly for large theta0, so AGM is preferred.
With L=1.0 m and g=9.81 m/s^2, at theta0=90 deg we get K(sin 45 deg)=1.8541, T=4*0.3193*1.8541=2.369 s and a 18.1% error. At theta0=175 deg, k=sin(87.5 deg)~0.9990, K~5.435, T~6.943 s and a 246% error (the period is roughly 3.5 times T0). At theta0->180 deg, K(1) diverges and the period becomes infinite: reaching the unstable upright equilibrium requires infinite time. In phase space this limit is the separatrix that divides oscillation orbits from rotation orbits.
For small theta0 (below about 30 deg) the series T = T0*(1 + theta0^2/16 + 11*theta0^4/3072 + 173*theta0^6/737280 + ...) reaches 0.01% accuracy in only one or two terms. At theta0=45 deg the two-term expansion gives T/T0 ~ 1.0386, virtually identical to the exact value. For theta0 above 90 deg the series converges slowly and many terms are needed, so the AGM-based elliptic integral is faster and more reliable. This tool uses AGM throughout, so the exact period is accurate from 1 deg up to 175 deg.