Rotational Motion
Lecturer:
Professor Stephen T. Thornton
Reading Quiz
Bonnie sits on the outer rim of a
merry-go-round, and Klyde sits
midway between the center and the
rim. The merry-go-round makes
one complete revolution every two
seconds.
Klyde’s angular velocity is:
A)
B)
C)
D)
E)
same as Bonnie’s
twice Bonnie’s
half of Bonnie’s
1/4 of Bonnie’s
four times
Bonnie’s
w
Klyde
Bonnie
Reading Quiz
Bonnie sits on the outer rim of a
merry-go-round, and Klyde sits
midway between the center and the
rim. The merry-go-round makes
one complete revolution every two
seconds.
Klyde’s angular velocity is:
The angular velocity w of any
point on a solid object
rotating about a fixed axis is
the same. Both Bonnie &
Klyde go around one
revolution (2p radians) every
two seconds.
A)
B)
C)
D)
E)
same as Bonnie’s
twice Bonnie’s
half of Bonnie’s
1/4 of Bonnie’s
four times
Bonnie’s
w
Klyde
Bonnie
Last Time
Collisions – elastic, inelastic, perfectly
inelastic
Center of mass
Changing mass - rockets
Today
Begin angular motion
Angular position, displacement
Angular speed, velocity
Angular acceleration
Similarities between translation and rotation
Pep Talk
Don’t get behind.
Halfway through course.
Starting most difficult part of course.
Rotational motion.
Six lectures
Angular momentum and torque are the most
difficult concepts of this course.
Define angular position, velocity,
and acceleration – just like we did
for translational motion.
 = angle
SI unit: radian (rad), dimensionless
One revolution = 3600 = 2π rad
Angular Position
 0
Arc Length
A rc length
s
 r
Sign Conventions
  0 counterclockw ise rotation
 < 0 clockw ise rotation
1 rev = 3600 = 2p rad
We will mostly use radians.
1 rad = 57.30
A radian is the angle for which the
arc length on a circle of radius r is
equal to the radius of the circle.
r
r
r
Angular Displacement
   f  i
angular displacem ent
Angular displacement and velocity
    f   i angular displacem ent
Just like with velocity, we divide angular
displacement by time to find angular
velocity.
Average angular velocity, wav
w av 

SI unit: radian per
second (rad/s) = s-1
t
w  lim
t  0

t

d
dt
Angular Speed and Velocity
w  0 counterclockw ise rotation
w < 0 clockw ise rotation
Vector characteristics
Magnitude of angular velocity is the angular speed.
But angular velocity is a vector.
Use right hand rule to obtain direction of angular
velocity.
Curl fingers in direction of rotation, and thumb
gives direction of angular velocity! (go back and
check). Vector direction is perpendicular to screen.
Conceptual Quiz:
You look at a bicycle as it moves from
your left to your right. The angular
velocity of the rear wheel is directed
A)
B)
C)
D)
E)
up
to the left
to the right
towards you
away from you
v
Answer: E – away from you
Use the right hand rule. The angular
velocity is into the screen and away
from you.
Angular acceleration
 av 
w
t
average angular acceleration
S I unit: rad/s  s
2
  lim
t  0
w
t
=
sam e S I unit s
dw
-2
instantaneous angular acceleration
dt
2
Vector direction follows right hand rule.
Angular Acceleration
A Pulley with Constant Angular Acceleration
 
dw

dt
 
w
t

w  w0
w  w0
t
 t  w  w0
w  w0   t
N ote: v  v 0  at
t0
Similarities between linear and
angular motion quantities ***
x
vw
a
w  w0   t
v  v0  a t
x  x0 
1
2
( v0  v )t
x  x0  v0t 
1
at
2
2
v  v  2 a ( x  x0 )
2
2
0
  0 
1
2
(w 0  w ) t
   0  w 0t 
w
2
w
2
0
1
t
2
2
 2  (   0 )
Angular Quantities
The frequency is the number of complete
revolutions or cycles per second:
f =
w
2p
cycles/s
Frequencies are measured in hertz:
1 Hz = 1 s
- 1
cycles/s
The period is the time one revolution takes:
T =
1
f
Linear to rotational quantities
T is period of one revolution.
w 
2 p rad
angular velocity around circle
T
vt 
2p r
T
 2p 
 r
  rw
 T 
tangential speed
vt  rw
Tangential speed depends
on radius.
Cooling Fan. A cooling fan is turned
off when it is running at 850 rev/min.
It turns 1350 revolutions before it
comes to a stop. (a) What was the
fan’s angular acceleration, assumed
constant? (b) How long did it take the
fan to come to a complete stop?
Conceptual Quiz
An object at rest begins to
rotate with a constant angular
acceleration. If this object has
angular velocity w at time t,
what was its angular velocity at
the time 1/2 t?
A)
B)
C)
D)
E)
1/2 w
1/4 w
3/4 w
2w
4w
Conceptual Quiz
An object at rest begins to
rotate with a constant angular
acceleration. If this object has
angular velocity w at time t,
what was its angular velocity at
the time 1/2t?
A)
B)
C)
D)
E)
1/2 w
1/4 w
3/4 w
2w
4w
The angular velocity is w = t (starting from rest),
and there is a linear dependence on time.
Therefore, in half the time, the object has
accelerated up to only half the speed.
Conceptual Quiz
An object at rest begins to rotate
with a constant angular
acceleration. If this object
rotates through an angle  in the
time t, through what angle did it
rotate in the time 1/2 t?
A)
B)
C)
D)
E)
/2
/4
3/4
2
4
Conceptual Quiz
An object at rest begins to rotate
with a constant angular
acceleration. If this object
rotates through an angle  in the
time t, through what angle did it
rotate in the time 1/2 t?
A)
B)
C)
D)
E)
/2
/4
3/4
2
4
The angular displacement is  = 1/2 t 2 (starting from
rest), and there is a quadratic dependence on time.
Therefore, in half the time, the object has rotated
through one-quarter the angle.
Centripetal and Tangential Acceleration
C entripetal acceleration
a cp 
v
2

( rw )
r
2
r
 r w  a cp
2
v
T angential acceleration
Not uniform
circular motion
vt  rw
 vt  r  w
at 
 vt
t
r
w
t
 r  a t
Conceptual Quiz
Bonnie sits on the outer rim of a
merry-go-round, and Klyde sits
midway between the center and the
rim. The merry-go-round makes one
revolution every 2 seconds. Who has
the larger linear (tangential) velocity?
A) Klyde
B) Bonnie
C) both the same
D) linear velocity is zero
for both of them
w
Klyde
Bonnie
Conceptual Quiz
Bonnie sits on the outer rim of a
merry-go-round, and Klyde sits
midway between the center and the
rim. The merry-go-round makes one
revolution every 2 seconds. Who has
the larger linear (tangential) velocity?
A) Klyde
B) Bonnie
C) both the same
D) linear velocity is zero
for both of them
Their linear speeds v will be
w
different because v = Rw and
Bonnie is located farther out
Klyde
(larger radius R) than Klyde.
VKlyde 
1
2
VBonnie
Follow-up: Who has the larger centripetal acceleration?
Bonnie
Do falling rigid body demo.