Rotational Kinematics

Chapter 8

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

A Brief Review from Chapter 5

Angular displacement:

Units: radians (rad)

S

rrS

A Brief Review from Chapter 5

Average angular

velocity:

units: rad/s

or: degrees/s, rev/min, etc.

x

r

v

r

v

t

Angular Acceleration

Average angular acceleration:

units: rad/s2

or: degrees/s2, rev/min2, etc.

0

0

ttt

Rotational Kinematic Equations

Definition of average angular velocity:

t

tt

2

12

1

0

0

Rotational Kinematic Equations

Definition of average angular acceleration:

t

tt

0

0

0

Rotational Kinematic Equations

A previous result:

2

0

2000

0

0

2

1

22

1

2

1

2

1

tt

tttt

t

t

Rotational Kinematic EquationsSolve definition of average acceleration for t:

Substitute into a previous result:

00

t

t

2

2

2

1

2

1

20

2

20

200

000

t

Comparison: Kinematic Equations Rotational Linear

( = constant) (a = constant)

2

2

12

1

20

2

20

0

0

tt

t

t

axvv

attvx

tvvx

atvv

2

2

12

1

20

2

20

0

0

Comparison: Kinematic EquationsSame equations, (some) different variables

Position, displacement: x

Time: t t

Velocity, speed: v

Acceleration: a

Average angular velocity is the angular displacement divided by the time interval in which it occurred.

x

r

v

rvr

v

t

rtv

rx

tvx

TT

T

T

) (small

Angular and Tangential Velocity

From the definition of linear acceleration:

From the definition of angular acceleration:

Combining:

Angular and Tangential Acceleration

t

rt

rr

t

vva TTT

000

t0

raT

From chapter 5: But:

Substituting:

Angular Velocity, Centripetal Acceleration

r

va TC

2

rvT

222

r

r

raC

2raC

The tangential and centripetal accelerations are vector components of the total acceleration.

Total Acceleration

C

T

TC

a

a

aaa

tan

22

When a circular, cylindrical, or spherical object rolls without slipping over a surface:

Rolling Motion: Velocity

rv linear speed of axle

wheel radius

angular speed of wheel

When a circular, cylindrical, or spherical object rolls without slipping over a surface:

Rolling Motion: Acceleration

ra linear acceleration of axle

wheel radius

angular acceleration of wheel

Angular displacement, , 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.

Angular Vectors

Angular velocity, , and angular acceleration, , are vectors.

Magnitudes: and

Directions: Parallel to the axis of rotation, and in the direction given by the right-hand rule:

Angular Vectors

t

t

Right-hand rule direction for :

Angular Vectors

Right-hand rule direction for :

Also parallel to axis of rotation Same direction as change in vector

Same direction as if is increasing in magnitude Opposite direction from if is decreasing in magnitude

Angular Vectors