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