Rotational Kinematics

1
Rotational Kinematics

• Chapter 8

2
Expectations

• After Chapter 8, students will
• understand and apply the rotational versions of
  the kinematic equations.
• be able to mathematically associate tangential
  variables with corresponding angular ones
• understand and apply the concept of total
  acceleration in rotational motion
• state and use the principle of rolling motion

3
A Brief Review from Chapter 5

• Angular displacement
• Units radians (rad)

4
A Brief Review from Chapter 5

• Average angular
• velocity
• units rad/s
• or degrees/s, rev/min, etc.

5
Angular Acceleration

• Average angular acceleration
• units rad/s2
• or degrees/s2, rev/min2, etc.

6
Rotational Kinematic Equations

• Definition of average angular velocity

7
Rotational Kinematic Equations

• Definition of average angular acceleration

8
Rotational Kinematic Equations

• A previous result

9
Rotational Kinematic Equations

• Solve definition of average acceleration for t
• Substitute into a previous result

10
Comparison Kinematic Equations

• Rotational
  Linear
• (a constant) (a
  constant)

11
Comparison Kinematic Equations

• Same equations, (some) different variables
• Position, displacement x q
• Time t t
• Velocity, speed v w
• Acceleration a a

12
Angular and Tangential Velocity
Average angular velocity is the angular
displacement divided by the time interval in
which it occurred.
13
Angular and Tangential Acceleration
From the definition of linear acceleration From
the definition of angular acceleration Combini
ng
14
Angular Velocity, Centripetal Acceleration
From chapter 5 But
Substituting
15
Total Acceleration
The tangential and centripetal accelerations are
vector components of the total acceleration.
16
Rolling Motion Velocity
When a circular, cylindrical, or spherical object
rolls without slipping over a surface
linear speed of axle
angular speed of wheel
wheel radius
17
Rolling Motion Acceleration
When a circular, cylindrical, or spherical object
rolls without slipping over a surface
linear acceleration of axle
angular acceleration of wheel
wheel radius
18
Angular Vectors
Angular displacement, q, is not a vector
quantity. the reason addition of angular
displacements is not commutative. Where you end
up depends on the order in which the angular
displacements (rotations) occur.
19
Angular Vectors
Angular velocity, w, and angular acceleration, a,
are vectors. Magnitudes and
Directions Parallel to the axis of rotation,
and in the direction given by the right-hand rule
20
Angular Vectors
Right-hand rule direction for w
21
Angular Vectors

• Right-hand rule direction for a
• Also parallel to axis of rotation
• Same direction as change in w vector
• Same direction as w if w is increasing in
  magnitude
• Opposite direction from w if w is decreasing in
  magnitude