Transcript Lecture 18.Collision..

Collisions & Center of Mass
Lecturer:
Professor Stephen T. Thornton
Reading Quiz
The disk shown below in (1)
clearly has its center of mass
at the center. Suppose the disk
is cut in half and the pieces
arranged as shown in (2).
Where is the center of mass of
(2) as compared to (1) ?
A)
B)
C)
D)
(1)
X
CM
higher
lower
at the same place
there is no
definable CM in
this case
(2)
Reading Quiz
The disk shown below in (1)
clearly has its center of mass at
the center. Suppose the disk is
cut in half and the pieces
arranged as shown in (2).
Where is the center of mass of
(2) as compared to (1) ?
The CM of each half is closer
to the top of the semicircle
than the bottom. The CM of
the whole system is located
at the midpoint of the two
semicircle CMs, which is
higher than the yellow line.
A) higher
B) lower
C) at the same
place
D) there is no
definable CM in
this case
(1)
X
CM
(2)
CM
Last Time
Defined linear momentum
Relationship between K.E. and momentum
More general form of 2nd law
Impulse
Internal and external forces
Today
Collisions – elastic, inelastic, perfectly
inelastic
Center of mass
Changing mass - rockets
Collisions
When two or more objects strike each other, we
have a collision.
Consider for now only two
objects. Let the net external
force = 0. Internal forces are
not considered.
Momentum is conserved. ***
In general, kinetic energy is not
conserved. ***
Kinds of Collisions
1 m v 2  1 m v 2  ? 1 m v' 2  1 m v' 2
2 A A 2 B B 2 A A 2 B B
Elastic collision – kinetic energy (K or KE)
is conserved. (definition of elastic collision.)
Inelastic collision – KE not conserved due to
lost energy (heat, sound, deformation, etc.)
Perfectly inelastic collision – objects
coalesce together.
Do demo with air track cars.
Remember that linear momentum is
always conserved in all collisions with no
external force – elastic or inelastic.
• One dimensional collisions are easiest
to consider.
• In two dimensions, linear momentum is
conserved in both directions. It is a
vector.
Collisions in Two or Three Dimensions
Conservation of energy and momentum can also be used to
analyze collisions in two or three dimensions, but unless
the situation is very simple, the math quickly becomes
unwieldy.
Here, a moving object collides with an object initially at
rest. Knowing the masses and initial velocities is not
enough; we need to know the angles as well in order to
find the final velocities.
Copyright © 2009 Pearson Education, Inc.
pi  p f
pi  m1v1,i  p f  (m1  m2 )v f
Conceptual Quiz:
After a collision between a large truck and a
small car, the impulse given to the truck by the
car is
A) larger than that given to the car by the
truck.
B) smaller than that given to the car by the
truck.
C) equal to that given to the car by the truck.
D) dependent upon the collision being elastic
or inelastic.
Answer: C - they are equal
Remember that impulse is equal
to the change in linear
momentum. There are no
external forces. Linear
momentum is conserved.
An Elastic Collision Between Two Air Carts
m1  m2
Elastic Collisions Between Air
Carts of Various Masses
Equal mass
Elastic Collisions Between Air
Carts of Various Masses
m2
m1
Elastic Collisions Between
Air Carts of Various Masses
m1
m2
Relative velocities in head-on
elastic collisions are equal in
magnitude but opposite in
direction. Used often in
problems. Easier than
conserving kinetic energy.
v A  vB  (v A ' vB ')
Projectile Explosion. A 224-kg projectile,
fired with a speed of 116 m/s at a 60.0° angle,
breaks into three pieces of equal mass at the
highest point of its arc (where its velocity is
horizontal). Two of the fragments move with the
same speed right after the explosion as the
entire projectile had just before the explosion;
one of these moves vertically downward and the
other horizontally. Determine (a) the velocity of
the third fragment immediately after the
explosion and (b) the energy released in the
explosion.
Center of Mass
The center of mass is special because the
object acts in many cases as if all the
mass of an object is concentrated at that
point.
The center of mass of an object is the
point where the system is balanced in a
uniform gravitational field.
The Center of Mass of Two Objects
No rotation!
Center of Mass of Two Objects
X CM
m1 x1  m2 x2

m1  m2
X CM
m1 x1  m2 x2 1


x
dm
M
M
Note that M  m1  m2
The center of mass is clearly nearest
the heavier mass.
Center of mass of three point-like objects:
It does not matter
where you place
your coordinate
system.
rCM
m1r1  m2 r2  ...  mi ri
R

m1  m2  ...
M
Center of Mass of the Arm
It does not matter where you
place your coordinate
system.
x
For an extended object, we imagine making it up of
tiny particles, each of tiny mass, and adding up the
product of each particle’s mass with its position and
dividing by the total mass. In the limit that the
particles become infinitely small, this gives:
rCM
m r


rCM
i i
M
1

r dm

M
Copyright © 2009 Pearson Education, Inc.
The center of gravity is the point at which the
gravitational force can be considered to act.
It is the same as the center of mass as long as
the gravitational force does not vary among
different parts of the object.
Copyright © 2009 Pearson Education, Inc.
Do center of mass demos
Center of mass of Virginia
Toys
Motion of Center of Mass
Velocity of center of mass:
m1v1  m2v2  ...  mi vi
VCM 

m1  m2  ...
M
Acceleration of center of mass:
m1a1  m2 a2  ...  mi ai
ACM 

m1  m2  ...
M
Newton’s 2nd Law for System
MACM  Fnet,ext
The CM of a system of particles with
total mass M moves as if M is acted
upon by the net, external force.
More on Center of Mass Motion
Note if there is no net external force,
then the acceleration of the center of
mass is zero.
MAcm  Fnet,ext  0
More on Center of Mass Motion
MAcm  Fnet,ext
If there is a net
external force, the
CM moves as if
all the net force
were acting at the
center of mass.











Crash of Air Carts
Note motion of CM.
Center of Mass of an
Exploding Rocket
Conceptual Quiz:
The object is suspended from two points
as shown. Which number lies closest to
the center of mass?
Use A, B, C… for 1, 2, 3…
Answer: D
#4
Look and see which number is always
on a straight vertical line from the
point of hanging. It helps to draw a
line.
Ballistic
Pendulum
2
v0
M m

 2 gh 

 m 
L  x
2

gx
L
2
 M m


 m 

L
x
Do ballistic pendulum demo and calculate v0.
m = 2.00 g
M = 3.81 kg
L = 4.00 m
2
(9.81)
x
2
v0 
4.00












2
3.81 0.002  0.89107 x2
0.002
2
The general motion of an object can be
considered as the sum of the translational
motion of the CM, plus rotational, vibrational,
or other forms of motion about the CM.
Systems of Variable Mass; Rocket Propulsion
Applying Newton’s second law to the
system shown gives:
dP = ( M + dM )(v + dv ) - (Mv + dMu )
å
dP
dv
dM
Fext =
= M
- (u - v )
dt
dt
dt
or
dv
M
=
dt
å
dM
Fext + vrel
dt
where vrel = u  v
Conceptual Quiz
A box slides with initial velocity 10 m/s
A) 10 m/s
on a frictionless surface and collides
B) 20 m/s
inelastically with an identical box. The
C) 0 m/s
boxes stick together after the collision.
D) 15 m/s
What is the final velocity?
E) 5 m/s
vi
M
M
M
M
vf
Conceptual Quiz
A box slides with initial velocity 10 m/s
A) 10 m/s
on a frictionless surface and collides
B) 20 m/s
inelastically with an identical box. The
C) 0 m/s
boxes stick together after the collision.
D) 15 m/s
What is the final velocity?
E) 5 m/s
The initial momentum is:
M vi = (10) M
vi
M
M
The final momentum must be the same!!
The final momentum is:
Mtot vf = (2M) vf = (2M) (5)
M
M
vf
Conceptual Quiz
A) the car
A small car and a large truck
collide head-on and stick
together. Which one has the
larger momentum change?
B) the truck
C) they both have the same
momentum change
D) can’t tell without knowing the
final velocities
Conceptual Quiz
A) the car
A small car and a large truck
collide head-on and stick
together. Which one has the
larger momentum change?
B) the truck
C) they both have the same
momentum change
D) can’t tell without knowing the
final velocities
Because the total momentum of the
system is conserved, that means that
p = 0 for the car and truck combined.
Therefore, pcar must be equal and
opposite to that of the truck (–ptruck) in
order for the total momentum change
to be zero. Note that this conclusion
also follows from Newton’s Third Law.
Follow-up: Which one feels
the larger acceleration?
Conceptual Quiz
A uranium nucleus (at
rest) undergoes fission
and splits into two
fragments, one heavy
and the other light.
Which fragment has the
greater momentum?
A) the heavy one
B) the light one
C) both have the same
momentum
D) impossible to say
1
2
Conceptual Quiz
A uranium nucleus (at
rest) undergoes fission
and splits into two
fragments, one heavy
and the other light.
Which fragment has the
greater momentum?
A) the heavy one
B) the light one
C) both have the same
momentum
D) impossible to say
The initial momentum of the uranium
was zero, so the final total momentum
of the two fragments must also be zero.
Thus the individual momenta are equal
in magnitude and opposite in direction.
1
2
Conceptual Quiz
If all three collisions
below are totally
inelastic, which one(s)
will bring the car on the
left to a complete halt?
A)
B)
C)
D)
E)
only I
only II
I and II
II and III
all three
Conceptual Quiz
If all three collisions
below are totally
inelastic, which one(s)
will bring the car on the
left to a complete halt?
In case I, the solid wall
clearly stops the car.
In cases II and III, because
ptot = 0 before the collision,
then ptot must also be zero
after the collision, which
means that the car comes
to a halt in all three cases.
A)
B)
C)
D)
E)
I
II
I and II
II and III
all three