Classical Mechanics
Unit 13 Examples
Unit 14 Concepts
Today’s Examples:
Today’s Concepts:
a) Elastic collisions…
a) Rotational Motion
b) Impulse, Force
during collisions
b) Moment of Inertia
Mechanics Lecture 14, Slide 1
Main Points
Mechanics Lecture 13, Slide 2
Main Points
Mechanics Lecture 13, Slide 3
Main Points
Mechanics Lecture 13, Slide 4
Kinetic Energy of the System
Mechanics Lecture 13, Slide 5
Kinetic Energy of the System
Mechanics Lecture 13, Slide 6
Kinetic Energy of the System
Mechanics Lecture 13, Slide 7
Kinetic Energy of the System
Mechanics Lecture 13, Slide 8
Kinetic Energy of the System
Mechanics Lecture 13, Slide 9
Mechanics Lecture 13, Slide 10
Kinetic Energy of the System
Mechanics Lecture 13, Slide 11
Kinetic Energy of the System
Mechanics Lecture 13, Slide 12
HW Problem
Mechanics Lecture 13, Slide 13
pi
q
q
pf
Another way to look at it:
DP = pf -pi
-pi
pf
|p|=mv
|DPX | = 2mv cosq
|DPY | = 0
Mechanics Lecture 13, Slide 14
|Favg | = |DP | /Dt = 2mv cosq /Dt
Mechanics Lecture 13, Slide 15
pi
pf
Another way to look at it:
DP = pf -pi
pf
-pi
Favg = DP/Dt
|DP| = |pf –pi| = m|vf –vi|
Dt = DP/Favg
1 2 1 2
DK = mv f - mvi
2
2
Mechanics Lecture 13, Slide 16
Ball Hits Wall 2
|p|=mv
Mechanics Lecture 14, Slide 17
Ball Hits Wall 2
|DPX | = 2mv cosq
|DPY | = 0
DP = DPx2  DPy2 = DPx
Dt = DP/Favg
Mechanics Lecture 14, Slide 18
Ball Hits Wall 2
DP = pf -pi
pf
-pi
|DP| = |pf –pi| = m|vf –vi|
Mechanics Lecture 14, Slide 19
Ball Hits Wall 2
Favg = DP/Dt
1 2 1 2
DK = mv f - mvi
2
2
Mechanics Lecture 14, Slide 20
Explosion
Ptot = 0
Mechanics Lecture 14, Slide 21
Explosion
m1  m2  m3 = mtotal
m3 = mtotal - m1 - m2
Ptot , x = m1v1 cos(180- q1 )  m2v2 cos(270 q 2 )  m3v3 cosq3 = 0
v3, x = v3 cosq3 =
- m1v1 cos(180- q1 )  m2v2 cos(270 q 2 ) 
m3
Ptot , y = m1v1 sin(180- q1 )  m2v2 sin(270 q 2 )  m3v3 sin q3 = 0
v3, y = v3 sin q3 =
- m1v1 sin(180- q1 )  m2v2 sin(270 q 2 ) 
m3
Mechanics Lecture 14, Slide 22
Explosion
vcm = Ptot / mtotal
3
1 2
DK = K f - K i = K f =  mi vi
i =1 2
1
1
1
DK = m1v12  m2 v22  m3v32
2
2
2
v32 = v32, x  v32, y
Mechanics Lecture 14, Slide 23
Explosion 2
m1v1 = 3m1v2
m1  3m1 = mtotal
m1 =
mtotal
4
 
ptot = p1  p2 = 0
1
1
m1v12  3m1 v22 = E
2
2
2
mv 
1
1
m1v12  3m1  1 1  = E
2
2
 3m1 
2

1


m
1
2
1
  = E
m  3m1 
v1
2 1 2
 3m1  

v1 =
E
2 
 m1 
3 
Mechanics Lecture 14, Slide 24
Explosion 2
m1v1 = 3m1v2
v2 =
Favg , 2 =
m1v1 v1
=
3m1 3
Dp2 p2 m2 v2
=
=
Dt
Dt
Dt
Dp1 = m1v1
vCM
ptotal
=
mtotal
Mechanics Lecture 14, Slide 25
Classical Mechanics
Unit 14 Concepts
Today’s Concepts:
a) Rotational Motion
b) Moment of Inertia
Mechanics Lecture 14, Slide 26
Lecture Thoughts
Mechanics Lecture 14, Slide 27
Unit 14 Main Points
Mechanics Lecture 14, Slide 28
Unit 14 Main Points
Mechanics Lecture 14, Slide 29
Unit 14 Main Points
Mechanics Lecture 14, Slide 30
Linear Dynamics
Describes motion of
point particles (or
center of mass)…
Mechanics Lecture 14, Slide 31
Rotational Kinematics
Needed to
fully describe
motion of
systems of
particles…or
extended
objects
Motion relative to center of mass
Mechanics Lecture 14, Slide 32
Rotational Kinematics
Mechanics Lecture 14, Slide 33
Comparison
Mechanics Lecture 14, Slide 34
Comparison for constant acceleration
Mechanics Lecture 14, Slide 35
Motion of point at distance R from Center
Mechanics Lecture 14, Slide 36
Motion of point at distance R from Center
Mechanics Lecture 14, Slide 37
Lecture Question
A.
B.
C.
0%
0%
0%
Mechanics Lecture 14, Slide 38
Summary of Rotations
Angular velocity w is measured in radians/sec
Frequency f is measured in revolutions/sec
1 revolution = 2p radians
Period T = 1/f
2p
w=
T
Mechanics Lecture 14, Slide 39
Another Summary
dw
a=
dt
at = Ra
v = wR
v2
ac = = w 2 R
R
w
Constant a does not mean constant w
Mechanics Lecture 14, Slide 40
Clicker Question
A.
B.
C.
A disk spins at 2 revolutions/sec.
0%
0%
0%
What is its period?
A) T = 2 sec
B) T = 2p sec
C) T = ½ sec
Mechanics Lecture 14, Slide 41
Clicker Question
A.
B.
C.
A disk spins at 2 revolutions/sec.
0%
0%
0%
What is its angular velocity?
A) w = 2p
B) w =
p
2
C) w = 4p
rad/sec
rad/sec
rad/sec
Mechanics Lecture 14, Slide 42
CheckPoint
A wheel which is initially at rest starts to turn with a constant angular
acceleration. After 4 seconds it has made 4 complete revolutions.
How many revolutions has it made after 8 seconds?
A) 8
B) 12
C) 16
a
Everyone got this right!!!
Mechanics Lecture 14, Slide 43
CheckPoint Response
After 4 seconds it has made 4 complete revolutions.
How many revolutions has it made after 8 seconds?
A) 8
B) 12
C) 16
a
C) The number of revolutions is proportional to time squared.
Mechanics Lecture 14, Slide 44