The Simple Pendulum

For small swings, a pendulum keeps time regardless of how far it swings.

12 minUpdated 21 October 2026simple pendulumangular SHMsmall-angle approximationmeasuring g

Lesson at a glanceShow the key points

Angular SHM: a restoring torque τ = −kθ gives θ = θ₀ sin(ωt + δ) with ω = √(k/I) and T = 2π√(I/k) (12.15–12.18).
A simple pendulum feels a restoring torque mgl sin θ; for small θ, sin θ ≈ θ and the motion is angular SHM.
Period T = 2π√(l/g) (12.23), independent of mass and (for small swings) of amplitude.
At large amplitudes the true period is longer: about 1.7% at 30°, 18% at 90°.
Lab method: time many oscillations, measure the effective length to the bob's centre, and use g = 4π²l/T².

Galileo, so the story goes, timed a swinging cathedral lamp with his pulse and noticed that big and small swings took the same time. The pendulum became the world’s timekeeper for three centuries.

Angular SHM

A body free to turn about an axis (a hanging umbrella nudged aside, say) can oscillate in angle. We call it angular simple harmonic motion when there is a position θ = 0 of zero torque, and a displacement θ produces a torque proportional to θ that tries to bring the body back:

\Gamma = -k\theta \quad\Rightarrow\quad \alpha = \frac{\Gamma}{I} = -\frac kI\theta = -\omega^2\theta \qquad (12.15)

This is the SHM equation with θ in place of x, so everything carries over:

Angular SHM(12.16–12.20)

\theta = \theta_0\sin(\omega t + \delta), \quad \omega = \sqrt{k/I}, \quad T = 2\pi\sqrt{I/k}, \quad E = \tfrac12k\theta_0^2

θ₀ is the angular amplitude; potential energy ½kθ², kinetic energy ½I(dθ/dt)²

Worked example

Writing an angular SHM

A body performs angular SHM with amplitude π/20 rad and period 0.40 s. At t = 0 it is at its maximum displacement. Write θ(t).

The simple pendulum

Definition · Simple pendulum

A heavy particle (bob) hung from a fixed point by a light, inextensible string: an idealisation of a small metal ball on a thread.

Let the string, of length l, make angle θ with the vertical. The tension passes through the pivot (no torque); the weight mg has lever arm l sin θ about the pivot and tries to restore θ = 0:

\Gamma = -mgl\sin\theta \qquad (12.21)

That isn’t proportional to θ, so a pendulum is not exactly SHM. But for small angles $\sin\theta \approx \theta$ (in radians), giving Γ = −mglθ. With $I = ml^2$ :

\alpha = -\frac{mgl}{ml^2}\theta = -\frac gl\theta

Period of a simple pendulum (small swings)(12.23)

T = 2\pi\sqrt{\frac lg}

independent of the bob’s mass and, for small amplitudes, of the amplitude

You get the same result treating the bob’s motion along its arc (x = lθ) as linear SHM: the tangential force is −mg sin θ ≈ −mg x/l, so $a = -(g/l)x$ .

Interactive

Pendulum lab

For small swings the period is 2π√(l/g), whatever the amplitude or the mass. Push the amplitude up and compare the true swing (solid) with the small-angle prediction (dashed): big swings take longer, so they fall behind.

T_0 = 2\pi\sqrt{l/g}

2.007s

true period2.022s

longer by0.77 %

World

Length

l

1.00m

Amplitude

\theta_0

20°

Worked example

A long pendulum

Find the period of a 1.5 m simple pendulum where g = 9.8 m/s². What length would beat seconds (T = 2.0 s)?

Measuring g with a pendulum

Hang a small ball from a clamp by a light thread. The effective length l runs from the point of suspension to the centre of the ball: thread length plus the ball’s radius (measured with slide callipers). Pull the ball slightly aside, release it, time 20 or 50 oscillations with a stopwatch, and divide to get T. Then $g = 4\pi^2l/T^2$ . Repeat for several lengths to reduce random errors.

Worked example

A school-lab measurement

With an effective length of 80.5 cm, a pendulum makes 25 oscillations in 45.0 s. Find g.

What to remember

1Angular SHM: $\Gamma = -k\theta$ , $T = 2\pi\sqrt{I/k}$ .
2Simple pendulum, small swings: $T = 2\pi\sqrt{l/g}$ , independent of mass and amplitude.
3Large swings take longer: the restoring torque grows only as sin θ.
4Measure g from g = 4π²l/T², timing many swings and using the length to the bob's centre.

Check yourself

A pendulum clock is taken to the Moon (g ≈ g_Earth/6). Its period becomes…

To double a pendulum's period you should make it…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

WatchPeriod of a pendulumKhan Academy PlayPendulum LabPhET · CU Boulder ReadPendulumsOpenStax University Physics §15.4 ReadPendulum motionThe Physics Classroom ReadSimple pendulumHyperPhysics ReadPendulum (mechanics)Wikipedia

PreviousEnergy in SHM Next Physical & Torsional Pendulums

The Simple Pendulum

For small swings, a pendulum keeps time regardless of how far it swings.

12 minUpdated 21 October 2026simple pendulumangular SHMsmall-angle approximationmeasuring g

Lesson at a glanceShow the key points

Angular SHM: a restoring torque τ = −kθ gives θ = θ₀ sin(ωt + δ) with ω = √(k/I) and T = 2π√(I/k) (12.15–12.18).
A simple pendulum feels a restoring torque mgl sin θ; for small θ, sin θ ≈ θ and the motion is angular SHM.
Period T = 2π√(l/g) (12.23), independent of mass and (for small swings) of amplitude.
At large amplitudes the true period is longer: about 1.7% at 30°, 18% at 90°.
Lab method: time many oscillations, measure the effective length to the bob's centre, and use g = 4π²l/T².

Galileo, so the story goes, timed a swinging cathedral lamp with his pulse and noticed that big and small swings took the same time. The pendulum became the world’s timekeeper for three centuries.

Angular SHM

\Gamma = -k\theta \quad\Rightarrow\quad \alpha = \frac{\Gamma}{I} = -\frac kI\theta = -\omega^2\theta \qquad (12.15)

This is the SHM equation with θ in place of x, so everything carries over:

Angular SHM(12.16–12.20)

\theta = \theta_0\sin(\omega t + \delta), \quad \omega = \sqrt{k/I}, \quad T = 2\pi\sqrt{I/k}, \quad E = \tfrac12k\theta_0^2

θ₀ is the angular amplitude; potential energy ½kθ², kinetic energy ½I(dθ/dt)²

Worked example

Writing an angular SHM

A body performs angular SHM with amplitude π/20 rad and period 0.40 s. At t = 0 it is at its maximum displacement. Write θ(t).

The simple pendulum

Definition · Simple pendulum

A heavy particle (bob) hung from a fixed point by a light, inextensible string: an idealisation of a small metal ball on a thread.

Let the string, of length l, make angle θ with the vertical. The tension passes through the pivot (no torque); the weight mg has lever arm l sin θ about the pivot and tries to restore θ = 0:

\Gamma = -mgl\sin\theta \qquad (12.21)

That isn’t proportional to θ, so a pendulum is not exactly SHM. But for small angles $\sin\theta \approx \theta$ (in radians), giving Γ = −mglθ. With $I = ml^2$ :

\alpha = -\frac{mgl}{ml^2}\theta = -\frac gl\theta

Period of a simple pendulum (small swings)(12.23)

T = 2\pi\sqrt{\frac lg}

independent of the bob’s mass and, for small amplitudes, of the amplitude

You get the same result treating the bob’s motion along its arc (x = lθ) as linear SHM: the tangential force is −mg sin θ ≈ −mg x/l, so $a = -(g/l)x$ .

Interactive

Pendulum lab

T_0 = 2\pi\sqrt{l/g}

2.007s

true period2.022s

longer by0.77 %

World

Length

l

1.00m

Amplitude

\theta_0

20°

Worked example

A long pendulum

Find the period of a 1.5 m simple pendulum where g = 9.8 m/s². What length would beat seconds (T = 2.0 s)?

Measuring g with a pendulum

Worked example

A school-lab measurement

With an effective length of 80.5 cm, a pendulum makes 25 oscillations in 45.0 s. Find g.

What to remember

1Angular SHM: $\Gamma = -k\theta$ , $T = 2\pi\sqrt{I/k}$ .
2Simple pendulum, small swings: $T = 2\pi\sqrt{l/g}$ , independent of mass and amplitude.
3Large swings take longer: the restoring torque grows only as sin θ.
4Measure g from g = 4π²l/T², timing many swings and using the length to the bob's centre.

Check yourself

A pendulum clock is taken to the Moon (g ≈ g_Earth/6). Its period becomes…

To double a pendulum's period you should make it…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

PreviousEnergy in SHM Next Physical & Torsional Pendulums

Angular SHM#

The simple pendulum#

Measuring g with a pendulum#

Check yourself

Keep exploring

Angular SHM#

The simple pendulum#

Measuring g with a pendulum#

Check yourself

Keep exploring

Angular SHM

The simple pendulum

Measuring g with a pendulum

Angular SHM

The simple pendulum

Measuring g with a pendulum