linear momentum final 1 edited

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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or,

ffii vmvmvmvm 22112211

Total momentum

before the

collision

Total momentum

after the

collision

fi PP


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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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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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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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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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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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Case 3: Decreasing Momentum over a

Short Time

Short impact times, the impact forces are

larger.


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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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26

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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28

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. 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


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

linear momentum final 1 edited

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