AP Physics C: Mechanics / Unit 5: Torque and Rotational Dynamics
5.4 Rotational Inertia
Notes
Rotational inertia $\displaystyle I=\sum mr^2$ measures resistance to changes in rotation. Mass farther from the axis increases $I$.
- Hoop: $MR^2$. Solid disk: $\displaystyle \frac{1}{2}MR^2$. Solid sphere: $\displaystyle \frac{2}{5}MR^2$.
- Parallel axis theorem: $I=I_{cm}+Md^2$.
Practice
Which has the largest $I$ for the same mass and radius?
All mass at the rim.
Two 1 kg masses at 2 m from the axis. $I=$
$2(1)(4)$.
Moving mass closer to the axis makes $I$:
Depends on $r^2$.
For a uniform rod about one end, $\displaystyle I=\int r^2dm=$
$\displaystyle \int_0^L\frac{M}{L}x^2dx$.
A uniform rod has $\displaystyle I_{cm}=\frac{1}{12}ML^2$. About one end (parallel axis):
$\displaystyle \frac{1}{12}+\frac{1}{4}$.
Ready for more? Take the Unit 5 test.