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2008 Physics 2111 Fundamentals of Physics
Chapter 9 1
Fundamentals of Physics
Chapter 9a Systems of Particles
1. A Special Point
2. The Center of Mass
3. Newton’s 2nd Law for a System of Particles
4. Linear Momentum
5. Linear Momentum of a System of Particles
6. Conservation of Linear Momentum
7. A Rocket
8. External Forces & Internal Energy Changes
Review & Summary
Questions
Exercises & Problems
2008 Physics 2111 Fundamentals of Physics
Chapter 9 2
The Center of Mass
• The center of mass of a body or system of bodies is the point that moves as though all of the mass were concentrated there and all external forces were applied there.
2008 Physics 2111 Fundamentals of Physics
Chapter 9 3
The Center of Mass
Consider 2 particles separated by distance d:
The center of mass, xcom, of the system is the point that moves as though all of the mass were concentrated there and all external forces were applied there.
xm x m x
m mcom
1 1 2 2
1 2
2008 Physics 2111 Fundamentals of Physics
Chapter 9 4
Center of Mass (cm)
comx
comx
xcom is closer to the larger mass.
xm
m mdcom
2
1 2
2008 Physics 2111 Fundamentals of Physics
Chapter 9 5
The Center of Mass
Many particles distributed along the x-axis:
The center of mass, xcom, of the system is the point that moves as though all of the mass were concentrated there and all external forces were applied there.
Many particles distributed in 3 dimensions:
i
n
iicom xm
Mx
1
1
i
n
iicom rm
Mr
1
1
i
n
iicom
i
n
iicom
i
n
iicom
zmM
z
ymM
y
xmM
x
1
1
1
1
1
1
2008 Physics 2111 Fundamentals of Physics
Chapter 9 6
Center of Mass?
20142803
16241803
15483
ii
ii
i
ym
xm
mM
1.115
16
i
iicom m
xmx 3.1
15
20
i
iicom m
ymy
2008 Physics 2111 Fundamentals of Physics
Chapter 9 7
CM of Solid Bodies
Uniform density case:
dmzM
z
dmyM
y
dmxM
x
com
com
com
1
1
1
dmrM
rcom 1
dVV
MdVdm
2008 Physics 2111 Fundamentals of Physics
Chapter 9 8
Newton’s 2nd Law for a System of Particles
r M m r
M r m r
M v m v
M a m a F F
com ii
n
i
com ii
n
i
com ii
n
i
com ii
n
i ii
n
net
11
1
1
1 1
comnet aMF
Net Force of all external forces
Internal forces cancel (Newton’s 3rd Law)
Total mass of the system
Acceleration of the center of mass
2008 Physics 2111 Fundamentals of Physics
Chapter 9 9
Newton’s 2nd Law for a System of Particles
The center of mass of a system moves like a particle of mass M under the influence of the net external force acting on the system.
• Net of all external forces
Internal forces cancel (Newton’s 3rd Law)
• Total mass of the system
• Acceleration of the center of mass
comnet aMF
2008 Physics 2111 Fundamentals of Physics
Chapter 9 10
Linear Momentum
Linear momentum of a particle:
The net force acting on a particle equals the time rate of change of the particle’s linear momentum:
dpdt
ddt
dvdtm v m ma
Fnet
dpdt
p mv
Newton’s 2nd Law
2008 Physics 2111 Fundamentals of Physics
Chapter 9 11
Linear Momentum of a System of Particles
The linear momentum of a system of particles equals the product of the total mass and the velocity of the center of mass:
Total linear momentum of a system of particles:
P p m vi
i
n
ii
n
i
1 1
But M v m vcom ii
n
i
1
P M vcom
Differentiating: com
comcom aM
dt
vdMvM
dt
d
dt
Pd
netFdt
Pd
2008 Physics 2111 Fundamentals of Physics
Chapter 9 12
Conservation of Momentum
Law of conservation of momentum: If no net external force acts on a system of particles, the total linear momentum of the system cannot change:
If the component of the net external force on a closed system is zero along an axis, then the component of linear momentum of the system along that axis cannot change.
netFdt
Pd
constantP
if PP
2008 Physics 2111 Fundamentals of Physics
Chapter 9 13
A shell explodes at its highest point.
M2
M
2M
the com continues on its way!
Coordinates of the explosion:
Momentum in the x-direction at the explosion:
0
20
21
21 2sin
g
vRxE
g
vyE 2
sin 200
000 2cos V
MvM
?maxx
g
yVxtVxx E
EE
200max Projectile motion from the explosion:
v0 = 20 m/s
= 600
2008 Physics 2111 Fundamentals of Physics
Chapter 9 14
Method 2: the com continues on its way!
2008 Physics 2111 Fundamentals of Physics
Chapter 9 15
The com continues on its way!
2008 Physics 2111 Fundamentals of Physics
Chapter 9 16
no friction “the system” = person + boat
When the person walks to the front of the boat, how far is the boat from the pier?
changenot does com theofmotion 0 netF
0restat initially is system the comv
afterxbeforex comcom
12060
312060
12060
5.31205.660
dd
md 5.2
?d
2008 Physics 2111 Fundamentals of Physics
Chapter 9 17
Conservation of ??
g
WM
g
wm
Initially both are moving with velocity v0; then the man starts to run with velocity vrel with respect to the car.
momentum before: 0vmM
momentum after: relvvmvM mM
vmvv rel
0
2008 Physics 2111 Fundamentals of Physics
Chapter 9 18
ROCKETS
2008 Physics 2111 Fundamentals of Physics
Chapter 9 19
Rockets
Further investigation and experimentation have confirmed the findings of Isaac Newton in the 17th century and it is now definitely established that a rocket can function in a vacuum as well as in an atmosphere. The Times regrets the error
• for after the rocket quits our air and really starts on its longer journey, its flight would be neither accelerated nor maintained by the explosion of the charges it then might have left. To claim that it would be is to deny a fundamental law of dynamics, and only Dr. Einstein and his chosen dozen, so few and fit, are licensed to do that.
…..That Professor Goddard, with his "chair" in Clark College and the countenancing of the Smithsonian Institution, does not know the relation of action to reaction, and of the need to have something better than a vacuum against which to react--to say that would be absurd. Of course he only seems to lack the knowledge ladled out daily in high schools.
NYT, January 13, 1920
NYT, July, 1969
2008 Physics 2111 Fundamentals of Physics
Chapter 9 20
Rockets
The rocket + exhaust gas has constant momentum in the absence of external forces (e.g. in outer space).
Constant Total Energy?
Constant Mechanical Energy?
Burning Fuel
Exhaust Gas
“The System”
Rocket
2008 Physics 2111 Fundamentals of Physics
Chapter 9 21
Fundamentals of Physics
Chapter 9b Collisions1. Collisions2. Impulse & Linear Momentum3. Momentum & Kinetic Energy4. Inelastic Collisions in 1-Dimension5. Elastic Collisions in 1-Dimension6. Collisions in 2-Dimensions
Review & SummaryQuestionsExercises & Problems
2008 Physics 2111 Fundamentals of Physics
Chapter 9 22
Collisions
A collision is an isolated event in which two or more bodies exert relatively strong forces on each other for a relatively short time.
A collision does not require physical contact.
Interacting
Moving freely
Moving freely
“Isolated” - no significant external forces during the collision.
2008 Physics 2111 Fundamentals of Physics
Chapter 9 23
Impulse During a Collision
Newton’s 2nd Law:
During a collision, the left & right bodies briefly exert a force on each other:F(t) acts for time t
An Impulse
Newton’s 3rd Law
dt
PdFnet
(eq. 9-23)
2008 Physics 2111 Fundamentals of Physics
Chapter 9 24
Impulse – Linear Momentum Theorem
(vectors!)
equal areas
Jppp if
dttFJ
tFJ avg
2008 Physics 2111 Fundamentals of Physics
Chapter 9 25
A Series of Collisions
A stream of objects (e.g. bullets, a stream of water, …) hitting a target:
What is the force on the target?
v depends on what happens to the objects in the stream? Stop, bounce back,…
J
p
Fm
tvavg
vmvmnpnJ
tFJ avg
2008 Physics 2111 Fundamentals of Physics
Chapter 9 26
Throw a ball against the wall J=? Favg = ?
a) Impulse on the ball?
b) Average force on the wall?
v v v v
v v v v vxf xi x
yf yi y
cos
sin sin
0
2
J p m v mv
J N s
2
18
sin
.
FJ
t
N s
sx Navg
18
0 0118 102.
..
030
01.010
0.6
3.0300
smst
v
kggm
sm
2008 Physics 2111 Fundamentals of Physics
Chapter 9 27
Momentum & Kinetic Energy in Collisions
Collisions in a closed, isolated system:– “Closed” - no mass enters or leaves– “Isolated” - no external forces
– Elastic Collision• Total Kinetic Energy is conserved.• Total Linear momentum is conserved.
– Inelastic Collision• Total Linear momentum is conserved.
• Completely Inelastic Collision– The bodies stick together after they collide.
The linear momentum of each colliding body may change but the total momentum of the system cannot change, whether the collision is elastic or inelastic.
2008 Physics 2111 Fundamentals of Physics
Chapter 9 28
Inelastic Collisions in One Dimension
afterbefore pp
ffii pppp 2121
ffii vmvmvmvm 22112211
momentum vectors:
2008 Physics 2111 Fundamentals of Physics
Chapter 9 29
Completely Inelastic Collisions
stuck together
ivmm
mV 1
21
1
afterbefore pp
fii ppp 1221
Vmmmvm i
21211 0
2008 Physics 2111 Fundamentals of Physics
Chapter 9 30
Velocity of the Center of mass
time
For an isolated system, the velocity of the center of mass cannot be changed by a collision:
Completely Inelastic Case:
vp p
m mcomi i
1 2
1 2
constant
ii ppP 21
comvMP
iicom vmvM
imM
2008 Physics 2111 Fundamentals of Physics
Chapter 9 31
The Ballistic Pendulum
Velocity of a speeding bullet?
•Step 1: Bullet quickly stops in block
Completely inelastic collision.
Little time for block to move.
(Tension in cords approximately perpendicular to block’s direction of travel; no work done.)
vMm
mV
hgMmVMm 221
•Step 2: Bullet & block swing upward
Mechanical Energy Conserved.
2008 Physics 2111 Fundamentals of Physics
Chapter 9 32
Colliding Blocks
Conservation of momentum: m v m v m v m vi i f f1 1 2 2 1 1 2 2
At maximum compression: v v vf f1 2
Conservation of total mechanical energy: K K Ui f spring
U k x 12
2
K m v m vi i i 12 1 1
2 12 2 2
2
K m m vf 12 1 2
2
algebra
xm m
k m mv vi i
1 2
1 21 2
m1 = 2.0 kg, 10 m/s
m2 = 5.0 kg, 3 m/s
k = 1120 N/m
2008 Physics 2111 Fundamentals of Physics
Chapter 9 33
Elastic Collisions in One Dimension
In an elastic collision, the kinetic energy of each colliding body may change, but the total kinetic energy of the system does not change.
m v m v m v
m v m v m v
i f f
i f f
1 1 1 1 2 2
12 1 1
2 12 1 1
2 12 2 2
2
vm m
m mvf i1
1 2
1 21
vm
m mvf i2
1
1 21
2
2008 Physics 2111 Fundamentals of Physics
Chapter 9 34
Elastic Collisions with a Moving Target
m v m v m v m vi i f f1 1 2 2 1 1 2 2
12 1 1
2 12 2 2
2 12 1 1
2 12 2 2
2m v m v m v m vi i f f
vm m
m mv
m
m mvf i i1
1 2
1 21
2
1 22
2
vm
m mv
m m
m mvf i i2
1
1 21
2 1
1 22
2
2008 Physics 2111 Fundamentals of Physics
Chapter 9 35
Planetary Assist (“slingshot”)
vm m
m mv
m
m mvf i i1
1 2
1 21
2
1 22
2
m m
m m
m m
m
m m
2 1
1 2
1 2
2
1 2
1
22
v v v
v
v
f i i
f
fkm
s
1 1 2
1
1
2
12 2 13
38
v f1v = 12 km/s
vJ= 13 km/s
2008 Physics 2111 Fundamentals of Physics
Chapter 9 36
Collisions in Two Dimensions
p p
p p p pbefore after
i i f f
1 2 1 2
For p m v m v m v
m v m vi f f
f f
21 1 1 1 1 1 2 2 2
1 1 1 2 2 2
0
0
: cos cos
sin sin
2008 Physics 2111 Fundamentals of Physics
Chapter 9 37
Billiard Ball Collision
m v m v m v
v v v
i f f
i f f
1 1 2
1 1 2
12 1
2 12 1
2 12 2
2
12
12
22
m v m v m v
v v v
i f f
i f f
Pythagorean Theorem!
Equal mass elastic scattering.
e.g. billiards
090
2008 Physics 2111 Fundamentals of Physics
Chapter 9 38
Equal Mass Elastic Collision
proton + proton proton + proton