Characterizing Circular Motion

•

Radius, r

•

Angular position,

θ

•

Angular displacement,

Δ

θ

•

Angular speed,

ω

=

Δ

θ

/

Δ

t

•

Angular acceleration,

α

=

Δ

ω

/

Δ

t

Example

What is the angular speed of a 33 1/3 rpm

record?

•

ω

=33.33 rev/min = 33.33 rev/60 sec

•

ω

= 33.33 ( 2

π

rad)/60 s = 3.49 rad/s

Acceleration in Circular Motion

Consider your rotating car tires as you

accelerate from 25 mph to 55 mph. What

is happening to the rotational speed of the

tires? R = 33 cm.

V = 11 m/s

V=24.5 m/s

ω

o

ω

f

Linear and Angular Connection

•

ω

= v/r

So,

ω

o

= (11 m/s)/0.33 m = 33 rad/s

And

ω

f

= (24.5 m/s)/0.33 m = 73.5 rad/s.

Therefore the change in angular speed,

ω

f

–

ω

o

= 40.5 rad/s =

Δ ω

Angular Acceleration

When you have changing angular speeds,

this means the object has an angular

acceleration,

α

(alpha),

which is calculated

by

•

α

=

Δ ω

/

Δ

t

In units of radians/second

2

= rad/s

2

Kinematics of Circular Motion

•

ω

=

Δ

θ

/

Δ

t

•

α

=

Δ

ω

/

Δ

t

•

ω

ave

=(

ω

f

+

ω

o

)/2

•

Θ

=

ω

ave

t

•

ω

f

=

ω

o

+

α

t

•

Θ

=

Θ

o

+

ω

o

t + ½

α

t

2

•

ω

f

2

=

ω

o

2

+ 2

αΔ

θ

Kinematics Example

A flywheel of a machine is rotating at 12

rev/s. Through what angle will the wheel

be displaced from its original position after

5 seconds?

•

Angular speed,

ω

= 12 rev/s = 75 rad/s

•

Θ

=

ω

ave

t = 75 rad/s * 5 s = 375 rad

•

= 214875

0

. 59.6875 revolutions, so .6875

revolutions from start position = 247

o

.

A turntable revolves at 33 1/3 rpm. It is shut

off and slow to a stop in 6.3 seconds.

What is the angular acceleration?

Through what angle did it turn as it slow to a

stop?

ω

f

=0,

ω

o

= 33.33 rpm =3.49 rad/s,

t = 6.3 s

•

ω

f

=

ω

o

+

α

t

•

Θ

=

Θ

o

+

ω

o

t + ½

α

t

2

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