Simple pendulum

The simple pendulum model

A simple pendulum consists of a point mass, called the bob, suspended from a light string of length $L$ and allowed to swing under gravity. The ideal model assumes the string has no mass, the pivot has no friction, air resistance is negligible, and the bob remains in a vertical plane.

The natural coordinate is the angle $heta$ from the vertical. The bob moves along an arc of radius $L$, so arc length is

$s=L heta$

when $heta$ is measured in radians.

Torque derivation

Gravity acts downward with magnitude $mg$. The component of gravity tangent to the arc is

$F_t=-mgsin heta$

The negative sign means the tangential force points back toward equilibrium at $heta=0$.

Using rotational dynamics about the pivot,

$au=Ialpha$

For a point mass at distance $L$,

$I=mL^2$

The torque from gravity is

$au=-mgLsin heta$

Since angular acceleration is

alpha=rac{d^2 heta}{dt^2}

we get

-mgLsin heta=mL^2rac{d^2 heta}{dt^2}

Cancel $mL$:

rac{d^2 heta}{dt^2}+rac{g}{L}sin heta=0

This is the exact ideal pendulum equation.

The small-angle approximation

The exact pendulum equation is nonlinear because of $sin heta$. For small angles measured in radians,

$sin hetaapprox heta$

This approximation is accurate when $heta$ is small, such as a few degrees. Then the equation becomes

rac{d^2 heta}{dt^2}+rac{g}{L} heta=0

This is the SHM equation with

omega=sqrt{rac{g}{L}}

Period of a small-angle pendulum

Using $T=2pi/omega$,

T=2pisqrt{rac{L}{g}}

The period depends on length and gravitational field strength, not on the mass of the bob. A longer pendulum swings more slowly. Stronger gravity makes it swing faster.

In the small-angle model, the period also does not depend on amplitude. For larger angles, the period increases slightly, and the simple formula becomes less accurate.

Energy of a pendulum

Choose gravitational potential energy zero at the lowest point. At angle $heta$, the bob is raised by height

$h=L(1-cos heta)$

so

$U=mgL(1-cos heta)$

The kinetic energy is

K=rac{1}{2}mv^2

Since $v=Ldot{ heta}$,

K=rac{1}{2}mL^2dot{ heta}^2

In the ideal model,

E=rac{1}{2}mL^2dot{ heta}^2+mgL(1-cos heta)

is conserved.

Pendulum versus mass-spring oscillator

A pendulum is not exactly a simple harmonic oscillator for all angles. Its restoring effect is proportional to $sin heta$, not $heta$. It becomes approximately SHM only when $sin hetaapprox heta$.

This is an important lesson: SHM often arises as a small-displacement approximation near stable equilibrium.

Physical applications

Pendulums have been used in clocks, seismometers, demonstrations of Earth's rotation, and gravimeters. They also provide an accessible example of how nonlinear systems can become linear under appropriate approximations.

The big idea

The simple pendulum is governed exactly by a nonlinear differential equation, but for small angles it becomes simple harmonic motion. Its period is approximately $2pisqrt{L/g}$, depending on length and gravity but not mass. The pendulum illustrates the power and limits of approximation in physics.