Physics 7B-1 (A/B)
Professor Cebra
Winter 2010
Lecture 8
Rotational Kinematics
and Angular
Momentum
Conservation
24-Feb-2010
Physics 7B Lecture 8
Slide 1 of 31
Rotational Motion
If the force is always perpendicular to the direction of the velocity, then only the
direction changes.
Centripetal Acceleration
Centripetal Force
v = 2pr/T
v vt

v
r
2
v v
ac 

t r
24-Feb-2010
Fc = mac=mv2/r
FC
Physics 7B Lecture 8
Slide 2 of 31
Angular Quantities
• There is an analogy between objects moving along a straight path
and objects moving along a circular path.
Linear
Position (x)
Velocity (v)
Acceleration (a)
Momentum (p)
Force (F)
Mass (m)
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Rotational
↔
↔
↔
↔
↔
↔
Angle (q)
Angular Velocity (w)
Angular Acceleration (a)
Angular Momentum (L)
Torque (t)
Moment of Inertia (I)
Physics 7B Lecture 8
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Angular Displacement
By convention,
q is measured
clockwise from
the x-axis.
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q
Physics 7B Lecture 8
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Angular Velocity
• Angular velocity is a vector:
– Right hand rule to determines direction of ω
– Velocity and radius determine magnitude of ω
r
vt
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vt
w
r
Physics 7B Lecture 8
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Angular Velocity
• What’s the angular velocity of a point
particle?
v
θ
r
r┴
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v
v
w 
r r sin q
Physics 7B Lecture 8
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Angular Acceleration
• Angular Acceleration: Rate of change in
angular velocity

 dw at
a

dt
r
• A stationary disk begins to rotate. After 3
seconds, it is rotating at 60 rad/sec. What is
the average angular acceleration?
w 60 rad/s  0 rad/s
rad
a

 20 2
t
3s
s
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Physics 7B Lecture 8
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Rotational Kinematics
Rotational Motion
(a = constant)
Linear Motion
(a = constant)
w  w0 + at
v = v0 + at
q = (1/2)(w0 + w)t
x = (1/2)(v0 + v)t
q = q0 + w0t + (1/2)at2
x = x0 + v0t + (1/2)at2
w2 = w02 + 2aq
v2 = v02 + 2ax
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Physics 7B Lecture 8
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Rotational Kinematics
A commercial jet airplane is idling at the end of the run way.
At idle, the turbo fans are rotating with an angular velocity of
175 rad/sec. The Tower gives the jetliner permission to takeoff. The pilot winds up the jet engines with an angular
acceleration of 175 rad/s2. The engines wind up through an
angular displacement of 2000 radians. What is the final
angular velocity of the turbo-fan blades?
Solution:
w2 = w02 + 2aq
= (175 rad/s)2 + 2(175 rad/s2)(2000 rad)
= 7.00 x 105 rad2/s2
w = 837 rad/s
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Physics 7B Lecture 8
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Angular Momentum
• What’s the momentum of a rolling disk?
v
• Two types of motion:


– Translation: p
 mdiskv


– Rotation:
L  I diskw
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Physics 7B Lecture 8
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Angular Momentum
• Angular Momentum: Product of position
vector and momentum vector
  
L  r  p  rp sin q
• Why is angular momentum important? Like
energy and momentum, angular momentum
is conserved.
• Angular Impulse: Change in angular momentum
  
vector
L  L2  L1
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Physics 7B Lecture 8
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Angular Momentum
• What’s the angular momentum of a point
particle?
p
θ
r
r┴
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L  r p  r sin q p
Physics 7B Lecture 8
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Torque
• Which way will the scale tip?
1 kg
1.5 kg
1 kg
1.5 kg
• Rotation of scale is influenced by:
– Magnitude of forces
– Location of forces
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Physics 7B Lecture 8
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Torque
• Which is more effective?
F1
F1
F2
• Rotation of wrench is influenced by:
– Magnitude of forces
– Location of forces
– Direction of forces
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Physics 7B Lecture 8
F2
DEMO: Torque Meter
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Torque
• Torque: The cause or agent of angular
acceleration   
t  r  F  rF sin q
• The angular velocity of an object will not change
unless acted upon by a torque

t Net on Object  0


w  0
• The net torque on an object is equal to the rate of
change of angular momentum 

t Net on Object
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dL

dt
Physics 7B Lecture 8
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Newton’s Second Law - Rotation

t Net on Object

dL dIw
dw


I
 Ia
dt
dt
dt
t = Ia
Angular impulse = tt = L
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Physics 7B Lecture 8
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Torque
• What’s the torque on a point particle?
F
θ
r
r┴
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t  r F  r sin q F
Physics 7B Lecture 8
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Rotational Kinetic Energy
• What’s the kinetic energy of a rolling disk?
v
• Two types of motion:
– Translation:
– Rotation:
24-Feb-2010
1
KEtran  mdiskv 2
2
1
KErot  I diskw 2
2
Physics 7B Lecture 8
Slide 18 of 31
Rolling Bodies
Angular Acceleration
t= I a
t = fR
Frictional Force:
f < msFN  msmg cosq
Linear Acceleration
Ftot = ma
Ftot = mg sinq – f
q
a = a/R
Normal Force:
FN = mg cosq
mg
Parallel Force:
FP = mg sinq
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Physics 7B Lecture 8
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Angular Acceleration
t= I a
t = fR
Moment of Inertia
Linear Acceleration
Ftot = ma
Ftot = mg sinq – f
a = a/R
ma  mg sin q  f
Ia
Ia
fR  Ia   f  2
R
R
Ia
ma  mg sin q  2
R
Ia
ma + 2  mg sin q
R
mR 2 + I a  mR 2 g sin q


mR 2 g sin q
a
mR 2 + I
24-Feb-2010
Demo: Crazy Rollers
As I increases,
a decreases
Physics 7B Lecture 8
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Moment of Inertia
Let’s use conservation of energy
1 2 1 2 1 2 1 v2
mgh  mv + Iw  mv + I 2
2
2
2
2 R
2
2
mR
gh
v2
mR 2 + I
vhoop  .707 2 gh
vdisk  .817 2 gh
24-Feb-2010
Physics 7B Lecture 8
Slide 21 of 31
Moment of Inertia
• Which will have the greater acceleration?
F=5N
F=5N
m2 = 2 kg
m1 = 1 kg
F=5N
F = ma
Fr = mra
t= mr2a
tIa
I  mR2
F=5N
m = 1 kg
m = 1 kg
• Which will have the greater angular
acceleration?
r1 = 1 m
24-Feb-2010
Physics 7B Lecture 8
r2 = 2 m
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Moment of Inertia
• What’s the moment of inertia of an extended
object?
I  I1 + I 2 + I 3 +   m1r1 + m2 r2 + m3r3 +    mi ri   r 2dm
2
1
I  MR 2
2
24-Feb-2010
2
1
I
ML2
12
Physics 7B Lecture 8
2
2
1
I  ML2
3
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Physics 7B Lecture 8
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Moment of Inertia
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Physics 7B Lecture 8
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Conservation of Angular Momentum
Conservation of L:
L = Iw
If moment of
inertia goes own,
then angular
velocity must go
up.
Demo: Rotation with Weights
24-Feb-2010
Physics 7B Lecture 8
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Conservation of Angular Momentum
Demo: Wheel
on a string
Demo: Gyro in a can
Demo: Gyroscope
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Physics 7B Lecture 8
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Conservation of Angular Momentum
The total angular momentum vector
must be conserved. If you flip the
spinning wheel, then something else
must create an angular momentum
vector.
DEMO: Spinning Wheel and rotation chair
24-Feb-2010
Physics 7B Lecture 8
Slide 28 of 31
Center of Gravity
Extended body
rotating with
constant angular
velocity
Parabolic trajectory
of the center of mass
The center of gravity (or Center of mass)
is the point about which an object will
rotate. For a body under goes both linear
projectile motion and rotational motion,
the location of the center of mass will
behave as a free projectile, while the
extended body rotates around the c.o.m.
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Physics 7B Lecture 8
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Center of Gravity
Parabolic
trajectory of
the center of
mass
Angular impulse
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Physics 7B Lecture 8
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Summary of Angular Equations
Magnitude of ω
v
w
r
KErot
Direction of ω
Same direction
as Δω
For point
particle
For extended
body
24-Feb-2010
1 2
 Iw
2
t  r F

w
a
t


t Net on Object
L  r P

L

t
For extended
body
Point particle
or extended
body
Same
direction
as ΔL


L  Iw
Physics 7B Lecture 8
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Announcements
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Physics 7B Lecture 8
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DL Sections
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Physics 7B Lecture 8
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Conservation of Angular Momentum
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Physics 7B Lecture 8
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Torque
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Physics 7B Lecture 8
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