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Academy
Simple Pendulums
Level 1 - Physics topic page in Oscillations.
Principle
A simple pendulum is harmonic only in the small-angle limit.
The motion is angular, but the restoring effect comes from the tangential component of gravity.
Notation
\(\theta\)
angular displacement from vertical
\(\mathrm{rad}\)
\(L\)
pendulum length
\(\mathrm{m}\)
\(g\)
gravitational field strength
\(\mathrm{m\,s^{-2}}\)
\(s\)
arc displacement
\(\mathrm{m}\)
\(\omega\)
small-angle angular frequency
\(\mathrm{rad\,s^{-1}}\)
\(T\)
period
\(\mathrm{s}\)
Method
Derivation 1: Relate angle to arc motion
The bob moves along a circular arc of fixed radius \(L\), so angular displacement determines tangential displacement.
Arc displacement
\[s=L\theta\]
Tangential acceleration
\[a_t=\frac{d^2s}{dt^2}=L\frac{d^2\theta}{dt^2}\]
Derivation 2: Resolve gravity tangentially
Tension is radial, so it does not restore the angle. The tangential component of weight points toward equilibrium.
Tangential force
\[F_t=-mg\sin\theta\]
Tangential Newton law
\[mL\frac{d^2\theta}{dt^2}=-mg\sin\theta\]
Cancel mass
\[\frac{d^2\theta}{dt^2}=-\frac{g}{L}\sin\theta\]
Derivation 3: Apply the small-angle limit
The equation becomes SHM only when the angle is small enough that \(\\sin\\theta\\approx\\theta\) in radians.
Small-angle approximation
\[\sin\theta\approx\theta\]
SHM equation
\[\frac{d^2\theta}{dt^2}=-\frac{g}{L}\theta\]
Angular frequency
\[\omega=\sqrt{\frac{g}{L}}\]
Period
\[T=2\pi\sqrt{\frac{L}{g}}\]
Rules
These are the compact results from the derivations above.
Arc displacement
\[s=L\theta\]
Small-angle equation
\[\frac{d^2\theta}{dt^2}=-\frac{g}{L}\theta\]
Pendulum frequency
\[\omega=\sqrt{\frac{g}{L}}\]
Pendulum period
\[T=2\pi\sqrt{\frac{L}{g}}\]
Examples
Question
A small-angle pendulum has length
\[0.80\,\mathrm{m}\]
Find its period using \[g=9.8\,\mathrm{m\,s^{-2}}\]
Answer
\[T=2\pi\sqrt{\frac{0.80}{9.8}}=1.79\,\mathrm{s}\]
Checks
- Angle must be in radians.
- Mass cancels for a simple pendulum.
- Larger length gives longer period.
- Large amplitudes are not exactly SHM.