Characterizing Circular Motion

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14 Νοε 2013 (πριν από 3 χρόνια και 10 μήνες)

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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