Chapter 8: Rotational Kinematics

8.1: Rotational Motion and Angular Displacement

Angular displacement: the angle at which a body rotates around a center or axis of rotation. Intersects axis of rotation perpendicularly.
- Equation: \Delta \theta =\theta -\theta _{0}
- SI units: radian \left( rad\right)
Counterclockwise is positive.
Clockwise is negative.
Angular displacement in radians = Arc length/ Radius
- Equation: \theta =\dfrac{s}{r}
- SI units: radian \left( rad\right)
Arc length equation: s=r\theta
- SI unit: meters
One full revolution: \theta =2\pi rad
Conversion: 1 rev = 2\pi rad=360^{\circ }

Average angular velocity = Angular displacement/Elapsed time
- Equation: \overline{\omega }=\dfrac{\Delta \theta }{\Delta t}
- SI units: rad/s
- Vector quantity
- Counterclockwise rotation is positive
- Clockwise rotation is negative
Average angular acceleration = Change in angular velocity/Elapsed time
- Equation: \overline{\alpha }=\dfrac{\Delta \omega }{\Delta t}
- SI units: rad/s^{2}
- Vector quantity
The angular acceleration is:
- Positive for counterclockwise rotations that are speeding up.
- Negative for counterclockwise rotations that are slowing down.
- Negative for clockwise rotations that are speeding up.
- Positive for clockwise rotations that are slowing down.

\alpha = constant

\omega =\omega _{0}+\alpha t

\theta =\dfrac{1}{2}\left( \omega _{0}+\omega \right) t

\theta =\omega _{0}t+\dfrac{1}{2}\alpha t^{2}

\omega ^{2}=\omega _{0}^{2}+2\alpha \theta

Tangential velocity: the linear speed of an object moving along a circular path at any given point. Radius x angular velocity.
- Equation: v_{T}=r\omega
- SI unit: m/s
- \omega must be in rad/s
- Vector quantity
- When there is no direction, it turns into a tangential speed
Tangential acceleration: how fast the tangential speed is changing.
- Equation: a_{T}=r\alpha
- SI unit: m/s^{2}
- \alpha must be in rad/s^{2}
Center of pivot has slowest tangential speed, outer has fastest tangential speed.

Centripetal acceleration: the rate of change of an object's tangential velocity when moving in a circular path.
- Equation: a_{c}=r\omega ^{2}
- SI unit: m/s^{2}
Tangential acceleration: how fast the tangential speed is changing.
- Equation: a_{T}=r\alpha
- SI unit: m/s^{2}
- \alpha must be in rad/s^{2}
Breaking down into x and y components: pythagorean theorem
- \overrightarrow{a}=\sqrt{\overrightarrow{a}_{c}^{2}+\overrightarrow{a_{T}}^{2}}

The tangential speed of a point on the outer edge of the tire is equal to the speed of the car over the ground.