PHY 113 A General Physics I
9-9:50 AM MWF Olin 101
Plan for Lecture 17:
Chapter 10 – rotational motion
1. Angular variables
2. Rotational energy
3. Moment of inertia
10/10/2012
PHY 113 A Fall 2012 -- Lecture 17
1
10/10/2012
PHY 113 A Fall 2012 -- Lecture 17
2
Angular motion
angular “displacement”  q(t)
dθ
ω
(t)

angular “velocity” 
dt dω
angular “acceleration”  α(t) 
dt
s
“natural” unit == 1 radian
Relation to linear
variables: sq = r (qf-qi)
vq = r w
aq = r a
10/10/2012
PHY 113 A Fall 2012 -- Lecture 17
3
v1=r1w
r1
w
r2
v2=r2w
Special case of constant angular acceleration: a = a0:
w(t) = wi + a0 t
q(t) = qi + wi t + ½ a0 t2
( w(t))2  wi2 + 2 a0 (q(t) - qi )
10/10/2012
PHY 113 A Fall 2012 -- Lecture 17
4
A wheel is initially rotating at a
rate of f=30 rev/sec.
R
What is the angular velocity?
ω  2πf  2 (30) rad/s
 188.495 rad/s
What is the speed of a dot on the rim
of the wheel at a radius R  0.5m?
v  wR  (188.495 rad/s)(0.5m)  94.247 m / s
10/10/2012
PHY 113 A Fall 2012 -- Lecture 17
5
A wheel is initially rotating at a
rate of f=30 rev/sec. Because of a
constant angular deceleration, the
wheel comes to rest in 3 seconds.
R
What is the angular deceleration?
0  2πf  2 (30) rad/s
a

3s
3s
 62.83 rad/s 2
What is the deceleration of a dot on the rim
of the wheel at a radius R  0.5m?
a  aR  (62.83 rad/s )(0.5m)  31.42m / s
2
10/10/2012
PHY 113 A Fall 2012 -- Lecture 17
6
2
Example: Compact disc motion
w1
w2
In a compact disk, each spot on the disk passes the laser-lens
system at a constant linear speed of vq = 1.3 m/s.
w1=vq/r1=56.5 rad/s
w2=vq/r2=22.4 rad/s
What is the average angular acceleration of the CD over the time
interval Dt=4473 s as the spot moves from the inner to outer radii?
a = (w2-w1)/Dt =-0.0076 rad/s2
10/10/2012
PHY 113 A Fall 2012 -- Lecture 17
7
Object rotating with constant angular velocity (a = 0)
w
R
v=Rw
v=0
Kinetic energy associated with rotation:
K   12mi vi2   12mi ri2ω2 
i
i
1
2
I
ω
;
2
where : I   mi ri2 “moment of inertia”
i
10/10/2012
PHY 113 A Fall 2012 -- Lecture 17
8
Moment of inertia:
I   mi ri
2
i
iclicker exercise:
A. a
10/10/2012
Which case has the larger I?
B. b
PHY 113 A Fall 2012 -- Lecture 17
9
Moment of inertia:
I   mi ri
2
i
I  2Ma
10/10/2012
2
I  2Ma  2mb
2
PHY 113 A Fall 2012 -- Lecture 17
2
10
Note that the moment of inertia depends on
both
a) The position of the rotational axis
b) The direction of rotation
m
m
d
d
I=2md2
10/10/2012
m
m
d
d
I=m(2d)2=4md2
PHY 113 A Fall 2012 -- Lecture 17
11
iclicker question:
Suppose each of the following objects each has the same total
mass M and outer radius R and each is rotating counterclockwise at an constant angular velocity of w=3 rad/s. Which
object has the greater kinetic energy?
(a) (Solid disk)
10/10/2012
(b) (circular ring)
PHY 113 A Fall 2012 -- Lecture 17
12
Various moments of inertia:
R
R
R
solid cylinder:
I=1/2 MR2
10/10/2012
solid sphere:
solid rod:
I=2/5 MR2
I=1/3 MR2
PHY 113 A Fall 2012 -- Lecture 17
13
Calculation of moment of inertia:
Example -- moment of inertia of solid rod through an axis
perpendicular rod and passing through center:
R
M
M R 2
1


2
I   mi ri    dr r     r dr  MR 2
3
i
 2 R  R
R  2R 
2
10/10/2012
R
PHY 113 A Fall 2012 -- Lecture 17
14
Note that any solid object has 3 moments of inertia;
some times two or more can be equal
j
i
k
iclicker exercise:
Which moment of inertia is the smallest?
(A) i
(B) j
(C) k
10/10/2012
PHY 113 A Fall 2012 -- Lecture 17
15
iclicker exercise:
Three round balls, each having a mass M and
radius R, start from rest at the top of the incline.
After they are released, they roll without slipping
down the incline. Which ball will reach the bottom
first?
B
A
C
I A  MR 2
1
I B  MR 2  0.5MR 2
2
2
I C  MR 2  0.4 MR 2
5
10/10/2012
PHY 113 A Fall 2012 -- Lecture 17
16
Total kinetic energy of
rolling object :
K total  K rolling  K CM
1 2 1
2
 Iw  MvCM
2
2
Note that :
dq
w
dt
ds
dq
R
 Rw  vCM
dt
dt
10/10/2012
K total  K rolling  K CM
1 I
1
2
2
(Rw )  MvCM

2
2R
2
1 I
 2
  2  M vCM
2 R

PHY 113 A Fall 2012 -- Lecture 17
17
iclicker exercise:
Three round balls, each having a mass M and
radius R, start from rest at the top of the incline.
After they are released, they roll without slipping
down the incline. Which ball will reach the bottom
first?
B
A
C
I A  MR 2
1
I B  MR 2  0.5MR 2
2
2
I C  MR 2  0.4 MR 2
5
10/10/2012
PHY 113 A Fall 2012 -- Lecture 17
18