AP Physics C: Mechanics / Unit 7: Oscillations
7.5 Simple and Physical Pendulums
Notes
- Simple pendulum (small angles): $\displaystyle T=2\pi\sqrt{\frac{L}{g}}$.
- Physical pendulum: a rigid object swinging about a pivot, $\displaystyle T=2\pi\sqrt{\frac{I}{mgd}}$, where $d$ is the pivot-to-center-of-mass distance.
- Both come from $\tau=-mgd\sin\theta\approx-mgd\,\theta$ for small angles.
Practice
A uniform rod of length $L$ swings from one end. $d=$
Center of mass at the middle.
The period of that rod ($\displaystyle I=\frac{1}{3}mL^2$):
$\displaystyle \frac{mL^2/3}{mgL/2}=\frac{2L}{3g}$.
The small-angle approximation uses:
Makes the motion simple harmonic.
Ready for more? Take the Unit 7 test.