AcademyOscillations
Academy
Physical Pendulums
Level 1 - Physics topic page in Oscillations.
Principle
A physical pendulum oscillates by gravitational torque about a pivot.
This extends the simple pendulum from a point mass on a light string to an extended rigid body.
Notation
\(\theta\)
angular displacement from equilibrium
\(\mathrm{rad}\)
\(I\)
moment of inertia about pivot
\(\mathrm{kg\,m^{2}}\)
\(m\)
body mass
\(\mathrm{kg}\)
\(d\)
pivot-to-center-of-gravity distance
\(\mathrm{m}\)
\(g\)
gravitational field strength
\(\mathrm{m\,s^{-2}}\)
\(T\)
small-angle period
\(\mathrm{s}\)
Method
Derivation 1: Build the torque model
The center of gravity is where the weight acts. Its lever arm about the pivot gives the restoring torque.
Weight torque
\[\tau=-mgd\sin\theta\]
Rotational dynamics
\[I\frac{d^2\theta}{dt^2}=\tau\]
Angular equation
\[I\frac{d^2\theta}{dt^2}=-mgd\sin\theta\]
Derivation 2: Take the small-angle limit
The physical pendulum becomes harmonic when the angular displacement is small enough for the sine approximation.
Approximate sine
\[\sin\theta\approx\theta\]
SHM equation
\[\frac{d^2\theta}{dt^2}=-\frac{mgd}{I}\theta\]
Angular frequency
\[\omega=\sqrt{\frac{mgd}{I}}\]
Period
\[T=2\pi\sqrt{\frac{I}{mgd}}\]
Derivation 3: Check the point-mass limit
A point mass at distance \(L\) from the pivot must reduce to the simple-pendulum result.
Point-mass inertia
\[I=mL^2\]
Center distance
\[d=L\]
Limit period
\[T=2\pi\sqrt{\frac{mL^2}{mgL}}=2\pi\sqrt{\frac{L}{g}}\]
Rules
These are the compact results from the derivations above.
Gravity torque
\[\tau=-mgd\sin\theta\]
Small-angle equation
\[\frac{d^2\theta}{dt^2}=-\frac{mgd}{I}\theta\]
Physical-pendulum period
\[T=2\pi\sqrt{\frac{I}{mgd}}\]
Point-mass limit
\[T=2\pi\sqrt{\frac{L}{g}}\]
Examples
Question
A uniform rod of length \(L\) pivots about one end. Use
\[I=\frac{1}{3}mL^2\]
and \[d=L/2\]
to find \(T\).Answer
\[T=2\pi\sqrt{\frac{I}{mgd}}=2\pi\sqrt{\frac{(1/3)mL^2}{mg(L/2)}}=2\pi\sqrt{\frac{2L}{3g}}\]
Checks
- Use moment of inertia about the pivot.
- Use center-of-gravity distance for torque.
- Small-angle approximation is still required.
- The pivot force has zero torque about the pivot.