Circular motion

clockwise is negative by convention

Angular Velocity & Acceleration

• $\vec{\omega }=\frac{d\vec{\theta }}{dt}$ is the angular velocity (in $rad/s$)
• $\vec{\alpha }=\frac{d\vec{\omega }}{dt}$ is the angular acceleration (in $rad/{\mathsf{s}}^2$)
• For a point $P$ at a distance $r$ from the axis of rotation
  • $\vec{\mathbf{a}}=\overrightarrow{{\mathbf{a}}_{\text{t}}}+\overrightarrow{{\mathbf{a}}_{\text{c}}}$
  • $a=\sqrt{a_{\text{t}}^2+a_{\text{c}}^2}$
    • $a_{\text{t}}=r\alpha$ is the tangential acceleration (in $m/{\mathsf{s}}^2$)
    • $a_{\text{c}}=\frac{v^2}{r}=\frac{r{\omega }^2}{r}=r{\omega }^2$ is the centripetal acceleration (in $m/{\mathsf{s}}^2$)
    • $v=r\omega$ is the tangential velocity (in $m/s$)
• For constant angular acceleration
  • $\theta ={\theta }_0+{\omega }_{0}t+\frac{1}{2}\alpha t^2$ is the angular position
  • $\omega ={\omega }_0+\alpha t$ is the angular velocity
  • ${\omega }^2={\omega }_0^2+2\alpha (\theta -{\theta }_0)$ is the angular velocity as a function of position

Moment of Inertia

• The moment of inertia (more accurately rotational inertia) of a rigid body relative to a given axis of rotation is defined as:
  • The ratio of the torque applied to the body to the resulting angular acceleration about the axis of rotation: $I=\frac{\tau }{\alpha }$
  • moment of inertia I = angular momentum L / angular velocity ω

Angular Momentum

• $\vec{\mathbf{L}}=\vec{\mathbf{r}}\times \vec{\mathbf{p}}$ is the angular momentum (or moment of momentum) vector (in $\mathsf{kg}\cdot {\mathsf{m}}^2/s$)
  • $\vec{\mathbf{r}}$ is the position vector (from the pivot point to the point of application of the force) (in $\mathsf{m}$)
  • $\vec{\mathbf{p}}$ is the momentum vector (in $\mathsf{kg}\cdot m/s$)
• $\vec{\mathbf{L}}=I\vec{\omega }$ is the angular momentum (in $\mathsf{kg}\cdot {\mathsf{m}}^2/s$)
  • $I$ is the moment of inertia (in $\mathsf{kg}\cdot {\mathsf{m}}^2$)
  • $\vec{\omega }$ is the angular velocity (in $rad/s$)
• todo
  • (Bozeman Science) Angular Momentum
  • (Bozeman Science) Conservation of Angular Momentum
  • Giancoli, 8–8 Angular Momentum and Its Conservation

Torque (Moment of Force)

• The axis of rotation (or pivot point) $O$ is the point about which the object rotates

• $\vec{\tau }=\vec{\mathbf{r}}\times \vec{\mathbf{F}}$ is the torque or moment of force vector (in $\mathsf{N}\cdot \mathsf{m}$) (cross product)

• $\vec{\mathbf{r}}$ is the position vector (from the pivot point to the point of application of the force) (in $\mathsf{m}$)

• $\vec{\mathbf{F}}$ is the force vector (in $\mathsf{N}$)

• $\theta$ is the angle between $\vec{\mathbf{r}}$ and $\vec{\mathbf{F}}$

• $\tau ,r,F$ are the magnitudes of the vectors

• $\tau =rF\sin \theta =r{F}_{\perp }=r_{\perp }F$ is the magnitude of the torque

  • $F_{\perp }=F\sin \theta$ is the component of the force perpendicular to the position vector
  • The lever arm (or moment arm) $r_{\perp }=r\sin \theta =\frac{|\mathbf{r}\times \mathbf{F}|}{|\mathbf{F}|}$ is the perpendicular distance from the axis of rotation to the line along which the force acts

• $\alpha =\frac{\tau }{I}$ is the angular acceleration (in $rad/{\mathsf{s}}^2$)

• todo https://youtu.be/5Zrphnd_0VI?list=PLllVwaZQkS2rxqMXTH-cdE0LIX9Zi_oS1

Power

• $P=\tau \times \omega$ is the power (in $\mathsf{W}$)
  • $\tau$ is the torque (in $\mathsf{N}\cdot \mathsf{m}$)
  • $\omega$ is the angular velocity (in $rad/s$)