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8/10/2019 Linear Momentum Final 1 Edited
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Linear Momentum
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Enthusiasm is the energyand force that builds literal
momentum of the humansoul and mind.
- Bryant H. McGill
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Linear Momentum
It is a measure of thedifficulty encountered
in bringing an object to
rest.A heavy and fast car is
harder to stop comparedto less heavy car with the
same speed.
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Linear Momentum
Property of an object related to its mass
and velocity. In equation; momentumpis
equal to mass mtimes velocity v
Momentum is a vector quantity
vmp
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Linear Momentum
Originally, Newtons 2ndLaw is stated in
terms of momentum:The rate of change of
momentum of a body is equal to the net force
applied to it.
A force is required to change the momentumof abody.
tpF /
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Conservation of Momentum
Consider twoparticles 1 and 2that collide on
each other andthereby exertingforce on each
other.
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Conservation of momentum
Assuming that the impulsive force isconstant, the change of momentum of a
particle 1 is
And the change in momentum of particle 2
due to the impulsive force of particle 1 is
)( 11112 ifon vvmtF
)( 22221 ifon vvmtF
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Conservation of momentum
So that the total impulse of the external
forces acting on the systemis just
From Newtons 3rdLaw,
thus
)()( 2221112112 ififonon vvmvvmtFF 02112 onon FF
ffii vmvmvmvm 22112211
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Conservation of Momentum
or,
ffii vmvmvmvm 22112211
Total momentum
before the
collision
Total momentum
after the
collision
fi PP
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Conservation of Momentum
In general,
If the vector sum of the external forces
on the system is zero, the totalmomentum of the system is constant
- Principle of Conservation ofMomentum
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12
The Principle of Conservation
of Linear Momentum
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13
Principle of Conservation of Linear Momentum:
The total linear momentum of an isolated system
remains constant(is conserved). An isolated systemis one for which the vector sum of the average
external forces acting on the system is zero.
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Impulse
Changes in momentum may occur when there iseither a change in the mass of an object,a change invelocity or both.
If momentum changes due to changing velocitywhile mass remains constant, accelerationoccurs.
Forceproduces the acceleration
For changing the momentum of an object, both forceand timeduring which the force acts areimportant.
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Impulse ImpulseIis the product of the net force Fand the t
time interval such force has in contact with the
object or
Impulse changes momentum in much the same way
that force changes velocity. Thus
Impulse = change in momentum
tFI
pI
The change in momentum of a body
du r ing a time interval equals the impu lse
of the net force that acts on the bod y
du r ing that interval .impu lse-momentum
theorem
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Impulse changes momentum
Case 1: Increasing momentum
If you wish to increase the momentum of
something as much as possible, you not
only apply the greatest forceyou can, you
also extend the time of application as
much as possible.
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Impulse changes momentum
Case 1: Increasing momentumLong-range cannons have long
barrels. The longer the barrel,
the greater the velocity of the
emerging cannonball. Why?The force of exploding
gunpowder in a long barrel acts
on the cannonball for a longer
time. This increased impulseproduces a greater momentum.
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Impulse changes momentum
Case 2: Decreasing Momentum over a
long time.
If you extend the time of impact 100 times,
impact force is reduced 100-fold. So
whenever you wish the force of impact to
be small, extend the time of impact.
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Case 2: Decreasing Momentum
over a long time.
A wrestler thrown to the floor tries to extend his
time of arrival on the floor by relaxing his
muscles and spreading the crash into a
series of impact as foot, knee, hip ribs, andshoulder fold onto the floor in turn. The
increased time of impact reduces the force of
impact.
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Case 2: Decreasing Momentum
over a long time. A person jumping from an
elevated position to a floor
below bends his knees
upon making contact,thereby extending the time
during which his
momentum is reduced 10
to 20 times that a stiff-legged, abrupt landing.
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Case 2: Decreasing Momentum
over a long time. Bungee jumping puts the
impulse-momentum
relationship to a thrilling
test. The long stretchingtime of the cord ensures a
small average force to
bring the jumper to a safe
halt before hitting theground.
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Impulse changes momentum
Case 3: Decreasing Momentum over a
Short Time
Short impact times, the impact forces are
larger.
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Case 3: Decreasing Momentum
over a Short Time The idea of short time of
contact explains how a
karate expert can sever a
stack of bricks with the blowof his bare hand. By swift
execution he makes the
time of contact very brief
and correspondingly makesthe force of impact huge.
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25
Impulse Example 1.
A Well-Hit Ball
A baseball (m=0.14kg) has initial velocity of v0=-
38m/s as it approaches a bat. The bat applies
an average force that is much larger than
the weight of the ball, and the ball departs
from the bat with a final velocity of vf=+58m.
(a)Determine the impulse applied to the ball by
the bat.
(b) Assuming time of contact is =1.6*10-3s, find
the average force exerted on the ball by the
bat.
F
t
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0mvmvJ f
)/38)(14.0()/58)(14.0( smkgsmkg
Ns
smkg
t
J
F 8400106.1
/.4.13
3
= +13.4 kg.m/s
(a)
(b)
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27
Example 2. A Rain Storm
Rain comes straight down with velocity of v0=-
15m/s and hits the roof of a car
perpendicularly. Mass of rain per second thatstrikes the car roof is 0.06kg/s.Assuming the rain
comes to rest upon
striking the car
(vf=0m/s), find the
average force exerted
by the raindrop.
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00 )( v
tm
t
mvmvF f
F = -(0.06kg/s)(-15m/s)=0.9
N
According to action-reaction law, the
force exerted on the roof also has a
magnitude of 0.9 N points downward: -
0.9N
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Collision
Football isn't a contact sport, it's a collision
sport. Dancing is a contact sport.
-Duffy Daugherty
A collisionis an isolated event in which two or
more moving bodies (colliding bodies) exert
forces on each other for a relatively short time.
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Elastic Collision
An elastic collision is anencounter between two
bodies in which the total
kinetic energy of the two
bodies after the encounter
is equal to their totalkinetic energy before the
encounter.
Elastic collisions are collisions
in which both momentum and
kinetic energy are conserved.
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Inelastic collision
in inelastic collision, the total kinetic energy
of the system is not conserved, however,
the momentum of the system is conserved.
Some of the kinetic energy before collision
is transformed into other types of energy.
The total kinetic energy after the collision is lessthan that before the collision.
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Inelastic collision
Completely inelastic collision
Colliding bodies stick together and move as
one body after collision.
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34
Assembling a Freight Train
Car 1 has a mass of m1=65*103kg and moves
at a velocity of v01=+0.8m/s. Car 2 has a
mass of m2=92*103
kg and a velocity ofv02=+1.3m/s. Neglecting friction, find the
common velocity vfof the cars after they
become coupled.
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(m1+m2) vf = m1v01 + m2v02
After collision Before collision
21
022011
mm
vmvmvf
)10921065(
)/3.1)(1092()/8.0)(1065(33
33
kgkg
smkgsmkg
=+1.1 m/s
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(m1+m2) vf = m1v01 + m2v02
After collision Before collision
21
022011
mm
vmvmvf
)10921065(
)/3.1)(1092()/8.0)(1065(33
33
kgkg
smkgsmkg
=+1.1 m/s
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37
A Collision in One Dimension
A ball of mass
m1=0.25kg and velocity
v01=5m/s collides head-on with a ball of mass
m2=0.8kg that is initially
at rest(v02=0m/s). No
external forces act on
the balls. If the collision
in elastic, what are the
velocities of the balls
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m1=0.25, m2=0.8
v01 =5 m/s, v02= 0
smvf
/62.258.025.0
8.025.0
1
smvf /38.258.025.0
25.022
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Collision between two objects of the same mass. One mass is at rest.
Collision between two objects. One not at rest initially has twice the mass.
Collision between two objects. One at rest initially has twice the mass.
Simple Examples of Head-On Collisions
(ELASTIC COLLISION : Energy and Momentum are Both Conserved)
vmp
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Collision between two objects of the same mass. One mass is at rest.
Collision between two objects. One not at rest initially has twice the mass.
Collision between two objects. One at rest initially has twice the mass.
Simple Examples of Head-On Collisions
(Totally Inelastic Collision, only Momentum Conserved)
vmp
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vmp
Collision between two objects of the same mass. One mass is at rest.
Example of Non-Head-On Collisions
(Energy and Momentum are Both Conserved)
If you vector add the total momentum after collision,
you get the total momentum before collision.
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A Collision in Two Dimensions
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A Collision in Two Dimensions A collision in two dimensions obeys the
same rules as a collision in one
dimension: Total momentum in each
direction is always the same before andafter the collision
Total kinetic energy is the same before
and after an elasticcollision
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Example 1Two objects slide over a
frictionless horizontal surface. The
first object, mass m1= 5kg , is
propelled with speed v1i= 4.5 m/s
toward the second object, mass
m2= 2.5 kg, which is initially atrest. After the collision, both
objects have velocities which are
directed = 30 on either side of
the original line of motion of the
first object. What are the final
speeds of the two objects? Is the
collision elastic or inelastic?
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Center of Mass The center of mass is the location where all of the mass of
the system could be considered to be located.
For a solid body it is often possible to replace the entire
mass of the body with a point mass equal to that of the
body's mass. This point mass is located at the center of
mass.
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Center of Mass For homogenous solid bodies that have a symmetrical
shape, the center of mass is at the center of body'ssymmetry, its geometrical center.
The center of mass is the point about which a solid will
freely rotate if it is not constrained.
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Center of Mass For a solid body the center of mass is also
the balance point. The body could besuspended from its center of mass and it
would not rotate, i.e. not be out of balance.
The center of mass of a solid body does not
have to lie within the body. The center of
mass of a hula-hoop is at its center wherethere is no hoop, just hula.
The center of mass for a system of
independently moving particles still has
meaning and is useful in analyzing the
interactions between the particles in thesystem.
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Center of Mass
In equation, it can be
summarized as
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Center of Mass
In equation, it can be
summarized as
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Center of Mass
For several bodies, the center of mass canbe obtained as
In three dimensions, it can be written as