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Chapter 9
Angular Motion in a Plane
ANGULAR DISPLACEMENT
• Angular displacement (q ) is usually
expressed in radians, in degrees, or
in revolutions.
One radian is the angle subtended at the center of a
circle by an arc equal in length to the radius of the
circle.
2
3
1
57.30
4
6 segments gets
to here.
2p segments gets
completely around.
6
5
1 rev = 3600 = 2p radians (rad)
Thus the angle q in radians is given in terms of
the arc length l it subtends on a circle of radius r
by
q  rl
The radian measure of an angle is a
dimensionless number.
THE ANGULAR SPEED
The angular speed (w ) of an object whose axis of
rotation is fixed is the rate at which its angular
coordinate, the angular displacement q, changes
with time. If q changes from qi to qf in a time t,
then the average angular speed is
w
q f  qi
t
• The units of w are exclusively rad/s. Since
each complete turn or cycle of a revolving
system carries it through 2p rad
w = 2p f.
• f is the frequency in revolutions per second,
rotations per second, or cycles per second.
• Accordingly, w is called the angular
frequency. We can associate a direction
with w and thereby create a vector quantity.
THE ANGULAR ACCELERATION
• The angular acceleration (a ) of an object whose
axis of rotation is fixed is the rate at which its
angular speed changes with time.
• If the angular speed changes uniformly from wi
to wf in the time t, then the angular acceleration
is constant and
a
w f  wi
t
a
w f  wi
t
The units of a are typically rad/s2, rev/min2, and such.
It is possible to associate a direction with w, and
therefore with a, thereby specifying the angular
acceleration vector a, but we will have no need to do
so here.
Equations for uniformly accelerated angular motion
are exactly analogous to those for uniformly
accelerated linear motion. In the usual notation we
have:
w
wi  w f
2
q  wt
w f  wi  a t
q  wi t  a t
1
2
2
w  w  2aq
2
f
2
i
RELATIONS BETWEEN ANGULAR
AND TANGENTIAL QUANTITIES:
• When a wheel of radius r rotates about an axis
whose direction is fixed, a point on the rim of
the wheel is described in terms of the
circumferential distance l it has moved, its
tangential speed v, and its tangential
acceleration aT.
• These quantities are related to the angular
quantities q, w, and a, which describe the
rotation of the wheel, through the relations:
l  rq
v  rw
aT  ra
l  rq
v  rw
aT  ra
• provided radian measure is used for q, w, and a.
v  rw
l  rq
aT  ra
Let’s derive these equations.

v
l
q
w
r
l  rq
v  rw
l  rq
v  r w
v  r w
t
t
aT  ra
l  r q
t
t
v  rw
If w gets bigger then v gets bigger.
aT is in the same direction of v.
If w gets smaller then v gets smaller.
aT is in the opposite direction of v.

v

v
q
w
r
l  rq
v  rw
aT  ra
• provided radian measure is used for q, w, and a.
• By simple reasoning, l can be shown to be the
length of belt wound on the wheel or the
distance the wheel would roll (without slipping)
if free to do so.
• In such cases, v and aT refer to the tangential
speed and acceleration of a point on the belt or
of the center of the wheel.
Cycloid
q
r
rq
Translation plus Rotation
Cycloid
Remember this slide
q
r
rq
What is happening to the velocity of this point?
ROTATIONAL INERTIA
• Law of inertia for rotating systems
An object rotating about an axis tends to remain
rotating at the same rate about the same axis unless
interfered with by some external influence.
• Examples: bullet, arrow, and earth
•
•
•
•
•
Demo – Football and spinning basketball
Demo - Whirly Tube (Zinger)
Demo – Whirly Shooter
Demo - Disc Gun
Demo - Rubber Bands
• Demo - Inertia Bars
• Moment of inertia (rotational inertia)
The sluggishness of an object to changes in
its state of rotational motion
• Distribution of mass is the key.
• Example: Tightrope walker
CENTRIPETAL ACCELERATION
• Centripetal acceleration (ac):
• A point mass m moving with constant speed v around a
circle of radius r is undergoing acceleration.
• The direction of the velocity is continually changing.
• This gives rise to an acceleration ac of the mass,
directed toward the center of the circle.
• We call this acceleration the centripetal acceleration;
its magnitude is given by
v
aC  r
2

v

v

v
q
w
r
Because v = rw, we also have
2 2
(
r
w
)
v
2
r
w
ac  r  r  r  rw
2
2
where w must be in rad/s.
THE CENTRIPETAL FORCE
The centripetal force (Fc) is the force that must act on
a mass m moving in a circular path of radius r to give
it the centripetal acceleration v2/r. From F = ma, we
have
FC  maC  m vr  m r rω  mrω2
2
2
2
Where FC is directed toward the center of the circular
path.
• Centripetal force - center seeking force
• Examples: tin can and string, sling, moon
and earth, car on circular path
• Demo - Coin on clothes hanger
• Demo - String, ball, and tube
• Demo - Loop the loop
CENTRIFUGAL FORCE
• Centrifugal force - center fleeing force
• Often confused with centripetal
• Examples: sling and bug in can
• Demo - Walk the Line
• Centrifugal force is attributed to inertia.
CENTRIFUGAL FORCE IN A
ROTATING REFERENCE FRAME
• A frame of reference can influence our view
of nature.
• For example: we observe a centrifugal
force in a rotating frame of reference, yet it
is a fictitious (pseudo) force.
• Centrifugal force stands alone (there is no
action-reaction pair) - it is a fictitious force.
• Another pseudo force - Coriolis