Chapter 10 Rotation of a Rigid Object About a Fixed Axis

Page	Description	Equation
	arc length	\[ s=r\theta \]
	radian-degree conversion	\[ \pi\ \mathrm{rad}=180^\circ \]
	angular displacement	\[ \Delta\theta=\theta_f-\theta_i \]
	average angular speed (scalar)	\[ \overline{\omega}=\frac{\Delta\theta}{\Delta t}\ \ \ \ \left(\frac{\mathrm{rad}}{\mathrm{s}}\right) \]
	(instantaneous) angular speed (scalar)	\[ \omega=\lim\limits_{\Delta t\rightarrow0}{\frac{d\theta}{dt}}\ \ \ \ \left(\frac{\mathrm{rad}}{\mathrm{s}}\right) \]
	sign of angular speed	\[ \begin{align*} &\omega>0 \text{ for counterclockwise,}\\ &\omega\lt 0 \text{ for clockwise rotation} \end{align*} \]
	average angular acceleration (scalar)	\[ \overline{\alpha}=\frac{\Delta\omega}{\Delta t}\ \ \ \ \left(\frac{\mathrm{rad}}{\mathrm{s}^2}\right) \]
	(instantaneous) angular acceleration (scalar)	\[ \alpha=\lim\limits_{\Delta t\rightarrow0}{\frac{d\omega}{dt}}\ \ \ \ \left(\frac{\mathrm{rad}}{\mathrm{s}^2}\right) \]
	sign of angular acceleration	\[ \begin{align*} &\alpha>0 \text{ for increasing,}\\ &\alpha\lt 0 \text{ for decreasing } \omega \end{align*} \]
	angular velocity (vector)	\[ \vect{\omega} \]
	angular acceleration (vector)	\[ \vect{\alpha} \]

Notes

direction of angular velocity, right-hand rule: along the axis out of the plane if $\omega\gt 0,$ along the axis into the plane if $\omega\lt 0.$

direction of angular acceleration, right-hand rule: parallel to $\omega$ out of the plane if $\omega$ is increasing, antiparallel to $\omega$ (into the plane) if $\omega$ is decreasing.

Let $\vect{P}$ and $\vect{Q}$ be any two points on a rigid object rotating about a fixed axis, $P$ and $Q$ be their distances from the axis, $\omega_P$ and $\omega_Q$ be their respective angular speeds, $\alpha_P$ and $\alpha_Q$ their respective angular accelerations, $v_P$ and $v_Q$ their respective linear speeds, and $a_P$ and $a_Q$ their respective linear accelerations. Then the following are true: \[ \begin{align*} &\omega_P=\omega_Q\\ &\alpha_P=\alpha_Q\\ &\text{If}\ P\gt Q \text{ then } v_P\gt v_Q\\ &\text{If}\ P\gt Q \text{ and } a_P\ne 0 \text{ or } a_Q\ne 0 \text{ then } a_P\gt a_Q \end{align*} \]