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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