PHYS 1443 – Section 001
Lecture #12
Wednesday, June 20, 2007
Dr. Jaehoon Yu
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Impulse and Linear Momentum Change
Collisions
Two Dimensional Collisions
Center of Mass
CM and the Center of Gravity
Remember the quiz tomorrow in class!!
Wednesday, June 20, 2007
PHYS 1443-001, Summer 2007
Dr. Jaehoon Yu
1
Collisions
Generalized collisions must cover not only the physical contact but also the collisions
without physical contact such as that of electromagnetic ones in a microscopic scale.
Consider a case of a collision
between a proton on a helium ion.
F
F12
t
F21
Using Newton’s
3rd
The collisions of these ions never involve
physical contact because the electromagnetic
repulsive force between these two become great
as they get closer causing a collision.
Assuming no external forces, the force
exerted on particle 1 by particle 2, F21,
changes the momentum of particle 1 by
r r
dp1  F21dt
Likewise for particle 2 by particle 1
r r
dp2  F12dt
law we obtain
dp2
r
 F12dt
So the momentum change of the system in the
collision is 0, and the momentum is conserved
Wednesday, June 20, 2007
r
  F21dt
 dp1
dp  dp1  dp2  0
psystem  p1  p2  constant
PHYS 1443-001, Summer 2007
Dr. Jaehoon Yu
2
Elastic and Inelastic Collisions
Momentum is conserved in any collisions as long as external forces are negligible.
Collisions are classified as elastic or inelastic based on whether the kinetic energy
is conserved, meaning whether it is the same before and after the collisions.
Elastic
Collision
A collision in which the total kinetic energy and momentum
are the same before and after the collision.
Inelastic
Collision
A collision in which the total kinetic energy is not the same
before and after the collision, but momentum is.
Two types of inelastic collisions:Perfectly inelastic and inelastic
Perfectly Inelastic: Two objects stick together after the collision,
moving together at a certain velocity.
Inelastic: Colliding objects do not stick together after the collision but
some kinetic energy is lost.
Note: Momentum is constant in all collisions but kinetic energy is only in elastic collisions.
Wednesday, June 20, 2007
PHYS 1443-001, Summer 2007
Dr. Jaehoon Yu
3
Elastic and Perfectly Inelastic Collisions
In perfectly Inelastic collisions, the objects stick
together after the collision, moving together.
Momentum is conserved in this collision, so the
final velocity of the stuck system is
How about elastic collisions?
m1v1i  m2v2i  (m1  m2 )v f
vf 
m1v1i  m2v2i
( m1  m2 )
m1v1i  m2v2i  m1v1 f  m2v2 f
1
1
1
1
In elastic collisions, both the
m1v12i  m2 v22i  m1v12f  m2 v22 f
2
2
2
2
momentum and the kinetic energy
m1 v12i  v12f   m2 v22i  v22 f 
are conserved. Therefore, the
final speeds in an elastic collision
m1 v1i  v1 f v1i  v1 f   m2 v2i  v2 f v2i  v2 f 
can be obtained in terms of initial From momentum
m1 v1i  v1 f   m2 v2i  v2 f 
speeds as
conservation above
 m  m2 
 2m2 
v1i  
v2i
v1 f   1
 m1  m2 
 m1  m2 
Wednesday, June 20, 2007
 2m1 
 m  m2 
v1i   1
v2i
v2 f  
 m1  m2 
 m1  m2 
PHYS 1443-001, Summer 2007
What happens when
the Dr.
two
masses are the same?
Jaehoon Yu
4
Example for Collisions
A car of mass 1800kg stopped at a traffic light is rear-ended by a 900kg car, and the
two become entangled. If the lighter car was moving at 20.0m/s before the collision
what is the velocity of the entangled cars after the collision?
The momenta before and after the collision are
Before collision
pi  m1v1i  m2v2i  0  m2v2i
m2
20.0m/s
m1
p f  m1v1 f  m2v2 f   m1  m2  v f
After collision
Since momentum of the system must be conserved
m2
vf
pi  p f
m1
vf
What can we learn from these equations
on the direction and magnitude of the
velocity before and after the collision?
Wednesday, June 20, 2007

 m1  m2  v f
 m2v2i
m2v2i
900  20.0i

 m1  m2  900  1800  6.67i m / s
The cars are moving in the same direction as the lighter
car’s original direction to conserve momentum.
The magnitude is inversely proportional to its own mass.
PHYS 1443-001, Summer 2007
Dr. Jaehoon Yu
5
Two dimensional Collisions
In two dimension, one needs to use components of momentum and
apply momentum conservation to solve physical problems.
m1
m1v1i  m2v2i  m1v1 f  m2v2 f
v1i
m2
q
f
x-comp.
m1v1ix  m2v2ix  m1v1 fx  m2v2 fx
y-comp.
m1v1iy  m2v2iy  m1v1 fy  m2v2 fy
Consider a system of two particle collisions and scattersin
two dimension as shown in the picture. (This is the case at
fixed target accelerator experiments.) The momentum
conservation tells us:
m1v1i  m2v2i  m1v1i
m1v1ix  m1v1 fx  m2 v2 fx  m1v1 f cos q  m2 v2 f cos f
m1v1iy  0  m1v1 fy  m2 v2 fy  m1v1 f sin q  m2 v2 f sin f
And for the elastic collisions, the
kinetic energy is conserved:
1
1
1
m1v 12i  m1v12f  m2 v22 f
2
2
2
Wednesday, June 20, 2007
PHYS 1443-001, Summer 2007
Dr. Jaehoon Yu
What do you think
we can learn from
these relationships?
6
Example for Two Dimensional Collisions
Proton #1 with a speed 3.50x105 m/s collides elastically with proton #2 initially at
rest. After the collision, proton #1 moves at an angle of 37o to the horizontal axis and
proton #2 deflects at an angle f to the same axis. Find the final speeds of the two
protons and the scattering angle of proton #2, f.
m1
v1i
Since both the particles are protons m1=m2=mp.
Using momentum conservation, one obtains
x-comp. m p v1i  m p v1 f cos q  m p v2 f cos f
m2
q
y-comp.
f
m p v1 f sin q  m p v2 f sin f  0
Canceling mp and put in all known quantities, one obtains
v1 f cos 37  v2 f cos f  3.50 105 (1)
From kinetic energy
conservation:
3.50 10 
5 2
 v v
2
1f
2
2f
v1 f sin 37  v2 f sin f
(2)
v1 f  2.80 105 m / s
Solving Eqs. 1-3
5
(3) equations, one gets v2 f  2.1110 m / s
Wednesday, June 20, 2007
f  53.0
PHYS 1443-001, Summer 2007
Dr. Jaehoon Yu
Do this at
home
7
Center of Mass
We’ve been solving physical problems treating objects as sizeless
points with masses, but in realistic situation objects have shapes
with masses distributed throughout the body.
Center of mass of a system is the average position of the system’s mass and
represents the motion of the system as if all the mass is on the point.
What does above
statement tell you
concerning forces being
exerted on the system?
m2
m1
x1
x2
xCM
Wednesday, June 20, 2007
The total external force exerted on the system of
total mass M causes the center of mass to move at
an acceleration given by a   F / M as if all
the mass of the system is concentrated on the
center of mass.
Consider a massless rod with two balls attached at either end.
The position of the center of mass of this system is
the mass averaged position of the system
m1 x1  m2 x2
CM is closer to the
xCM 
heavier object
m1  m2
PHYS 1443-001, Summer 2007
Dr. Jaehoon Yu
8
Motion of a Diver and the Center of Mass
Diver performs a simple dive.
The motion of the center of mass
follows a parabola since it is a
projectile motion.
Diver performs a complicated dive.
The motion of the center of mass
still follows the same parabola since
it still is a projectile motion.
Wednesday, June 20, 2007
The motion of the center of mass
of the diver is always the same.
PHYS 1443-001, Summer 2007
9
Dr. Jaehoon Yu
Example 9-12
Thee people of roughly equivalent mass M on a lightweight (air-filled)
banana boat sit along the x axis at positions x1=1.0m, x2=5.0m, and
x3=6.0m. Find the position of CM.
Using the formula
for CM
m x

m
i
xCM
i
i
i
i
M 1.0 M  5.0 M  6.0  12.0 M
 4.0(m)

3M
M M M
Wednesday, June 20, 2007
PHYS 1443-001, Summer 2007
Dr. Jaehoon Yu
10
Center of Mass of a Rigid Object
The formula for CM can be expanded to Rigid Object or a
system of many particles
xCM 
m1x1  m2 x2    mn xn
m1  m2    mn
m x

m
m y

m
i
i i
i
yCM
mi
ri
rCM
zCM
i
i
rCM  xCM i  yCM j  zCM k
rCM 
i
i

m x
i
i
i  mi yi j   mi zi k
i
i
m
i
i
m r
i
i i
i
M
A rigid body – an object with shape
and size with mass spread throughout
the body, ordinary objects – can be
considered as a group of particles with
mass mi densely spread throughout
the given shape of the object
Wednesday, June 20, 2007
i i
i
i
i
i
The position vector of the
center of mass of a many
particle system is
m z

m
i
xCM 
 m x
i
M
xCM  lim
m 0
PHYS 1443-001, Summer 2007
Dr. Jaehoon Yu
i
i
i
rCM 
 m x
i
i
M
i

1
rdm

M
1
M
 xdm
11
Example for Center of Mass in 2-D
A system consists of three particles as shown in the figure. Find the
position of the center of mass of this system.
Using the formula for CM for each
position vector component
y=2 m (0,2)
1
m x

m
i
(0.75,4)
xCM
rCM
i
One obtains rCM  x
CM
m x m x m x m x
xCM  m  m  m  m

i i
1 1
i
2 2
1
i
i
i
(2,0)
m3
x=2
(1,0)
m2
x=1
i
2
3 3
3

m2  2m3
m1  m2  m3
m y

m
i
yCM
i
i
i
i
r
r  m2  2m3  i  2m1 j
i  yCM j 
m1  m2  m3
If m1  2kg; m2  m3  1kg
i
m y
yCM  m

i
i
i
i
i
m y  m2 y2  m3 y3
2m1
 1 1

m1  m2  m3
m1  m2  m3
Wednesday, June 20, 2007
rCM
PHYS 1443-001, Summer 2007
Dr. Jaehoon Yu
3i  4 j

 0.75i  j
4
12
Example of Center of Mass; Rigid Body
Show that the center of mass of a rod of mass M and length L lies in midway
between its ends, assuming the rod has a uniform mass per unit length.
The formula for CM of a continuous object is
L
xCM 
x
dx
Therefore xCM
dm=ldx
1

M
1
M

xL
x 0
xdm
Since the density of the rod (l) is constant; l  M / L
The mass of a small segment dm  ldx
xL
1 1 2 1 1  L
1 1 2 

x0 lxdx  M  2 lx  M  2 lL   M  2 ML   2
x 0
xL
Find the CM when the density of the rod non-uniform but varies linearly as a function of x, la x
M

xL
x 0
ldx  
xL
x 0
axdx
xL
1
1

 aL2
  ax 2 
2
 x 0 2
Wednesday, June 20, 2007
xCM
1

M

xL
x 0
1
lxdx 
M
xL
1
a
x
dx

x 0
M
2
1 2
1 3
 2L
 aL  
 ML  
3
3
 M 3

PHYS 1443-001, Summer 2007
xCM 
1
M
Dr. Jaehoon Yu
xL
1 3 
 3 ax 

 x 0
13
Center of Mass and Center of Gravity
The center of mass of any symmetric object lies on an
axis of symmetry and on any plane of symmetry, if
object’s mass is evenly distributed throughout the body.
How do you think you can
determine the CM of
objects that are not
symmetric?
Center of Gravity
mi
Axis of
One can use gravity to locate CM.
symmetry
1. Hang the object by one point and draw a vertical line
following a plum-bob.
2. Hang the object by another point and do the same.
3. The point where the two lines meet is the CM.
Since a rigid object can be considered as a collection
of small masses, one can see the total gravitational
force exerted on the object as
Fg   Fi   mi g  Mg
i
mig
CM
What does this
equation tell you?
Wednesday, June 20, 2007
i
The net effect of these small gravitational
forces is equivalent to a single force acting on
a point (Center of Gravity) with mass M.
PHYS 1443-001, Summer 2007
14
The CoG is the point in an object
asDr.if Jaehoon
all theYugravitational force is acting on!
Motion of a Group of Particles
We’ve learned that the CM of a system can represent the motion of a system.
Therefore, for an isolated system of many particles in which the total mass
M is preserved, the velocity, total momentum, acceleration of the system are
d 1
drCM
 
dt  M
dt
Velocity of the system
vCM 
Total Momentum
of the system
pCM  MvCM
Acceleration of
the system
aCM
External force exerting
on the system
If net external force is 0
Wednesday, June 20, 2007

M
1

m
r

 i i  M
i i
i i
1

m
v

 i i  M
 Fext  MaCM   miai 
dptot
dt
dri
mi vi


dt
M
 m v  m v   p  p
M
d 1
dvCM
 
dt  M
dt
 Fext  0 
 mi
dptot
dt
ptot  const
PHYS 1443-001, Summer 2007
Dr. Jaehoon Yu
tot
 mi
dvi  mi ai

dt
M
What about the
internal forces?
System’s momentum
is conserved.
15