Dynamics of turning systems
Rotational dynamics connects the distribution of mass, the line of action of a force and the resulting angular motion. Angular momentum then explains why spinning systems resist changes to their rotational state.
Learning goals
- use angular displacement, velocity and acceleration;
- calculate torque and moment of inertia about a stated axis;
- apply $\sum\tau=I\alpha$ and rotational energy;
- analyse rolling without slipping;
- use angular momentum and external torque to explain conservation and precession.
1. Translational and rotational language
| Translation | Rotation about a fixed axis |
|---|---|
| position $x$, velocity $v$, acceleration $a$ | angle $\theta$, angular velocity $\omega$, angular acceleration $\alpha$ |
| mass $m$ | moment of inertia $I$ |
| force $F$ and $\sum F=ma$ | torque $\tau$ and $\sum\tau=I\alpha$ |
| $K=\tfrac12mv^2$, momentum $p=mv$ | $K_{rot}=\tfrac12I\omega^2$, angular momentum $L=I\omega$ |
For constant angular acceleration, $\omega=\omega_0+\alpha t$, $\theta=\omega_0t+\tfrac12\alpha t^2$ and $\omega^2=\omega_0^2+2\alpha\theta$. Radians must be used in these equations. Points on a rigid body share $\omega$, but their linear speeds differ: $v=\omega r$.
2. Torque: force with leverage
Torque is the vector $\boldsymbol\tau=\mathbf r\times\mathbf F$. Its magnitude is $\tau=rF\sin\phi=F\ell$, where $\ell$ is the perpendicular lever arm from the axis to the force line. The right-hand rule gives direction. A large force through the axis has zero torque; a smaller force far from the axis can turn an object strongly.