Chapter 8: Rotational Kinematics

8.1: Rotational Motion and Angular Displacement

Definition of 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 = $2\pi rad=360^{\circ }$

8.2: Angular Velocity and Angular Acceleration

• 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.

8.3: The Equations of Rotational Kinematics

The Equations of Kinematics for Rotational Motion

$\alpha$ = constant

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

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

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

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

Variables

8.4: Angular Variables and Tangential Variables

• 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.

8.5 Centripetal Acceleration and Tangential Acceleration

• 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

  • $\overrightarrow{a}=\sqrt{\overrightarrow{a}_{c}^{2}+\overrightarrow{a_{T}}^{2}}$

8.6 Rolling Motion

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

8.7: The Vector Nature of Angular Variable

• Right-Hand Rule