Test 2 Thursday, March 25 8:20 pm
Covers: Ch. 6, Ch. 7, Secs. 8.1-8.5
Bring: calculator, ID, formula sheet
(no phones, formula sheet 2-sided, handwritten)
Last year’s Exam 2 is now posted on our website
A-G
Room
TUR L007
H-M
CLB C130
N-U
WEIM 1064
V-Z
LIT 0113
1st letter last name
Kinetic Energy and Moment of Inertia

1
1
1
2
2
KEi  mi vi = mi  ri  =  mi ri 2   2
2
2
2
v
Moment of Inertia I
- measures rotational inertia of object
- depends on the axis
1 2
KE   KEi  I 
2
i
r
piece i of
rigid body,
mass mi
Kinetic Energy and Moment
of Inertia
I   mi ri
2
Compare to linear motion:
M   mi
i
1 2
KE  I 
2
(pure rotation)
1
2
KE  Mv
2
Rotational Kinetic Energy
1 2
KEr  I
2
Conservation of Mechanical Energy
(KEt  KEr  PE )i  (KEt  KEr  PE )f
Rolling and Kinetic Energy

P
1 2
KE  I  
2
 ...
1
1
2
 I cm  Mv 2
2
2
Instantaneous pure rotation about P
or
Combination of rotation about cm and translation of cm
KEt
KEr
Ball rolling down an incline
How fast does it leave the
bottom of the incline?
h
Conservation of Mechanical Energy
(KEt  KE r  PE )i  (KEt  KE r  PE )f
0
+ 0
+ mgh
= mv2/2+I2/2 + 0
I=2mr2/5 and r=v so I2/2= mv2/5
Giving: mgh = mv2/2+ mv2/5 = 7 mv2/10
Note m cancels out as usual
or
v = 10gh/7
angular momentum
REMINDERS
Linear Motion
Rotations
coordinate
velocity
acceleration
x
v
a
mass
force
1st Newton’s Law
2nd Newton’s Law
m
F
F=0: v=const
F = ma
Kinetic Energy
Momentum
K = ½mv2
p = ma



angle
angular velocity
angular acceleration


=0: =const
 = I
moment of inertia
torque
K = ½I2
???
Kinetic Energy
Angular Momentum
Torque and Angular Acceleration
Newton’s Second Law for a Rotating Object
  I
analogous to
∑F = ma
I = moment of inertia
2
2
I


m
r

MR
For Uniform Ring
i i
moment of inertia depends on
quantity of matter
and its distribution
and location of axis of rotation
Other Moments of Inertia
Angular Momentum of rigid body
L=Iω
Just like p = mv
L always conserved unless there is external torque!
Impulse
L
 
t

p
Just like  F 
t
Isolated system

Conservation of Angular
Momentum states: The angular
momentum of a system is
conserved when the net external
torque acting on the systems is
zero.
  0, Li  Lf or Ii i  If  f
Conservation of Angular
Momentum– Example 1
How does a skater
spin faster in the
air?
L is conserved
As arms come in-I decreases
 increases
http://www.youtube.com/watch?v=AQLtcEAG9v0
Applying Conservation Rules
In an isolated system, the
following
three quantities are
conserved:
 Mechanical energy
 Linear momentum
 Angular momentum
Example 2:
Angular momentum conservation
Gyroscope (for fun)
 
  r F
  rF , direction: 

L
pivot
point
dL   dt
dL  dt

d 
L
L
F
looking from above
)
L(t+dt
L(t)
dL
this gyroscope precesses
counterclockwise
d
Precession angular velocity:  
dt
 rF
 
L I