High SchoolExtendedPhysicsOscillationsEnergy

How does a pendulum work?

Q: What is the formula for the period of a pendulum?

T = 2\pi\sqrt{L/g}, valid for small swings (under about 15°). L is the string length in metres and g \approx 9.81\ \text{m/s}^2. A 1-metre pendulum has a period of about 2 seconds.

A pendulum's period depends only on its length and gravity, not its mass: T = 2π√(L/g). The physics of the swing, energy and damping, with a simulation.

How does a pendulum work?

A pendulum is a mass hanging from a fixed point. Pull it aside and gravity drags it back toward the lowest point; it arrives with speed, overshoots, and swings out the other side. For small swings the time of one full back-and-forth depends only on the length of the string and on gravity — $T = 2\pi\sqrt{L/g}$ — and not on the mass or on how far you pull it.

Key fact

For small angles a pendulum's period is

T = 2\pi\sqrt{L/g}

: longer string, slower swing; stronger gravity, faster swing; mass and amplitude do not matter.

Animated diagram showing a simple pendulum's back-and-forth motion with labeled components. — A simple pendulum: a point mass (the bob) on a string of length L, swinging through an angle θ from the vertical.

What does the period depend on?

Three quantities describe a swing: the amplitude (how far it swings), the period $T$ (time for one full cycle) and the frequency $f = 1/T$ (cycles per second). Three observations tell you almost everything about the period:

Longer string → slower swing. The period grows with the square root of the length, so doubling $L$ multiplies $T$ by only $\sqrt2 \approx 1.41$ .
Stronger gravity → faster swing. On the Moon, where $g$ is about six times smaller, the same pendulum swings roughly $2.4$ times slower.
Mass does not matter. Gravity pulls a heavier bob harder, but a heavier bob is also harder to accelerate — the two cancel exactly.

Why is it simple harmonic motion?

The bob feels two forces: the string's tension and gravity $mg$ . Tension is perpendicular to the motion, so only the component of gravity along the arc changes the speed. That tangential component always points back toward the bottom:

Free body diagram showing force vectors on a pendulum bob at maximum displacement. — Forces on the bob: tension along the string and gravity. The component of gravity along the arc, −mg sin θ, is the restoring force.

F_t = -mg\sin\theta \quad\Longrightarrow\quad \frac{d^2\theta}{dt^2} = -\frac{g}{L}\sin\theta

Pendulum equation of motion · formula card →

For small angles $\sin\theta \approx \theta$ (at $15°$ the error is only $0.4\%$ ), and the equation becomes the same one that governs a mass on a spring, with $\omega_0 = \sqrt{g/L}$ :

\frac{d^2\theta}{dt^2} = -\frac{g}{L}\,\theta \quad\Longrightarrow\quad \theta(t) = \theta_0\cos(\omega_0 t + \varphi), \qquad T = \frac{2\pi}{\omega_0} = 2\pi\sqrt{\frac{L}{g}}

Period of a simple pendulum · formula card →

Length $L$ (m)	Period $T$ (s)	Where you see it
0.25	1.00	Metronome, short playground swing
1.00	2.01	Standard laboratory pendulum
9.80	6.28	Large chandelier, church-bell scale

Measure gravity with a string

Time ten swings of a pendulum of known length and solve

g = 4\pi^2 L / T^2

. Geologists once used exactly this to map tiny changes in gravity across the Earth.

Where does the energy go?

Take the bottom of the swing as zero height. At angle $\theta$ the bob has risen $h = L(1-\cos\theta)$ , so its potential energy is $U = mgL(1-\cos\theta)$ and its kinetic energy is $K = \tfrac12 m v^2$ . At the extremes the bob stops and all the energy is potential; at the bottom it moves fastest and all of it is kinetic. Without friction the sum never changes:

E = \tfrac12 m L^2\dot\theta^{\,2} + mgL(1-\cos\theta) = mgL(1-\cos\theta_0)

What happens at large angles?

Without the small-angle shortcut the equation has no elementary solution — the exact period involves an elliptic integral. What matters in practice is the result: bigger swings take longer. The pendulum lags behind the $2\pi\sqrt{L/g}$ prediction more and more as the amplitude grows, and would take forever to swing up to exactly $180°$ (perfectly balanced upside-down).

Amplitude $\theta_0$	Period vs small-angle formula
15°	+0.4%
30°	+1.7%
60°	+7.3%
90°	+18%

Using energy conservation and the substitution

\sin(\phi/2) = k\sin\psi

with

k = \sin(\theta_0/2)

, the exact period is

T = 4\sqrt{L/g}\,K(k)

, where

K(k) = \int_0^{\pi/2} \dfrac{d\psi}{\sqrt{1-k^2\sin^2\psi}}

is the complete elliptic integral of the first kind. Expanding for small

\theta_0

gives

T \approx 2\pi\sqrt{L/g}\,\bigl(1 + \tfrac{1}{16}\theta_0^2 + \dots\bigr)

Why does a real pendulum stop?

Air resistance and friction at the pivot remove a little energy every swing, so the amplitude decays. Model that as a force proportional to velocity and there are three regimes: underdamped (it keeps swinging with shrinking amplitude), critically damped (it returns to rest as fast as possible without overshooting — what door closers aim for) and overdamped (it creeps back slowly). Real clocks fight damping by feeding the pendulum a small push each swing.

Beyond one pendulum

Hang a second pendulum from the first and the motion becomes chaotic — nearly identical starts diverge completely. Explore it in the Double Pendulum simulation, and see how the same math runs on a computer in the companion article below.

Frequently asked questions

What is the formula for the period of a pendulum?

T = 2\pi\sqrt{L/g}

, valid for small swings (under about 15°).

L

is the string length in metres and

g \approx 9.81\ \text{m/s}^2

. A 1-metre pendulum has a period of about 2 seconds.

Does the mass of the bob change the period?

No. A heavier bob feels a larger gravitational force but also resists acceleration more, and the two effects cancel exactly. Two bobs of different mass on strings of equal length swing together.

Does the amplitude change the period?

Only slightly for small swings — that near-independence is called isochronism and is what makes pendulum clocks work. At large amplitudes the period does grow: about 18% longer at 90°.

Why does a pendulum eventually stop?

Friction at the pivot and air resistance convert mechanical energy into heat a little at a time. Each swing is slightly smaller than the last until the motion dies out.

How do you make a pendulum swing faster?

Shorten the string. The period scales with the square root of the length, so quartering the length halves the period. Stronger gravity would also speed it up, but you cannot change that at home.

Keep exploring

Change length, gravity and amplitude in the Simple Pendulum simulation.
Watch chaos appear with two pendulums: Double Pendulum simulation.
How the same equations are solved on a computer: Simulating a pendulum in code.
The other classic harmonic oscillator: How a spring works.

How does a pendulum work?

How does a pendulum work?

What does the period depend on?

Why is it simple harmonic motion?

Where does the energy go?

What happens at large angles?

Why does a real pendulum stop?

Frequently asked questions

What is the formula for the period of a pendulum?

Does the mass of the bob change the period?

Does the amplitude change the period?

Why does a pendulum eventually stop?

How do you make a pendulum swing faster?

Keep exploring

Related Articles

What is physics? A visual introduction

Class 12 Physics Complete Guide – All Chapters with Formulas & Examples

Why is the sky blue?