Chapter 3 Force Impulse Momentum

8/18/2019 Chapter 3 Force Impulse Momentum

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Chapter 3: Dynamics :- Force, Impulse and Momentum

3.1 Force

What is a FORCE?

When we think of force

We usually imagine a push or a pull exerte on some o!"ect.

Exert a force on a !all when you throw it or kick it.

Exert a force on a chair when you sit own on it.

What happens to an o!"ect when it is acte on !y a force? #t epens on the magnitue an irection of the force.

Force is a $ector %uantity& thus ' we enote it with a irecte arrow' "ust as we o

for $elocity an acceleration.Early concept of Force (he early scientist introuce& force as the concept of a fiel .

(he corresponing forces are calle fiel force.

Example& when a mass' m is place at some point' ) near a secon mass' * 'we

say that m interacts with * !y $irtue of the gra$itational fiel that exists at ).

Funamental force (he known funamental force in nature are all fiel force.

(he funamental force are +

• strong nuclear force !etween su!atomic particles.

• electromagnetic force !etween electric charges at a rest or in motion.

• weak nuclear force' which arise in certain raioacti$e ecay processes

• an gra$itational force. (he force of gra$itational attraction !etween two o!"ects.

#,--C EW(O

1/0 2 1


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# o not know what # may appear to the worl' !ut to myself # seem to ha$e !een

only like a !oy playing on the seashore an i$erting myself in now an then fining a

smoother pe!!le or a prettier shell than orinary' whilst the great ocean of truth lay all

unisco$ere !efore me.

(he whole !uren of philosophy seems to consist in this4 from the phenomena

of motions to in$estigate the forces of nature an then from these forces to explain theirnature.

3.2 ewton5s 6aws of *otion

Newton’s First aw o! Motion

#nertia

Newton’s "econd aw o! Motion

*ass an Weight

Center of *ass an Center of 7ra$ity

Newton’s #hird aw o! Motion

ormal Force

Friction 2 ,tatic friction an 8inetic frictiont $elocity if there is no net external

force !etween the o!"ect an en$ironment.

ewton5s First 6aw of *otion

,tates+

9 -n o!"ect at rest stays at rest an an o!"ect in motion continues in motion with constan

#n e%uation+

:: =⇒=∑ a F

7arfiel testing

ewton5s First 6aw of *otion


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#nertia

;efinition + #nertia is the tenency of an o!"ect to resist any changes in its state of rest or

motion.

#n other wors +


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Real Experiences of #nertia


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)ull Force

)ush Force


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ewton5s secon 6aw of *otion

,tates+

• (he accelation of an o!"ect is irectly proportional to the net force acting on it an

in$ersely proportional to its mass.

am F

m F a

=⇒

=

∑∑

Restate+

• (he rate of change of momentum with time is proportional to the net applie force

an is in the same irection.

( ) ∑

∑

==

∆

∆=

∆

∆

∆∆

F am

t

vm

t

vm

F t

vmα

ote + (he irection of the acceleration is the same as the irection of the applie net

force.

,.#. unit of force + 1 ewton is the force that prouces an acceleration of 1 m s = when acting on a

1 kg mass. (herefore' 1 > 1kg m s=

*ass *ass is an inherent property of a !oy. *ass is a %uantitati$e measure of the inertia of a !oy. *ass is the force re%uire per unit of acceleration prouce. (he $alue of mass is inepenent of location. *ass is a scalar. ,.#. unit of mass is kilogram.

Weight Weight is the force exerte on an o!"ect !y a gra$itational fiel.

W > mg > Fg Weight $aries slightly with altitue !ecause weight epens on the strength of thegra$itational fiel.

gearth ≠ gmoon Weight is a $ector. ,.#. unit of weight is ewton or kg m s=.

Center of *ass


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;efination+

• (he point at which all the mass can !e consiere to !e


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,olution+

;efination+

• (he point at which all the mass can !e consiere to !e


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Examples + When you push on the wall it will push !ack with the same force. When little Bim pushes little (im' little (im pushes !ack with the same force. (he

!oy with the !etter grip on the groun will keep from falling. #n the game of tug of war' when one sie pull on the other sie' the other sie

pulls !ack with the same force.

ormal Force;efinition +

(he contact force exerte !y a surface onto a !oy resting or sliing on the

surface an acts perpenicularly to the surface.

θ

Friction Friction originates from forces !etween atoms an molecules when surfaces are in

contact. Example +

2 Friction occurs when a !oy mo$es on a rough surface or through a flui meiumwater' air' etc@. (he irection of the friction is parallel to the surface in contact an opposite to the

irection of in which an o!"ect wants to mo$e.

Friction is a retaring force that resists motion on a surface.

Nirection of

f motion

f N

f = N

is the coefficient of friction.


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is epens on the o!"ects in$ol$e an on the conition of the surface.

(wo (ypes of Friction

"tatic Friction

(inetic Friction

,tatic Friction ,tatic friction' f s is the force of friction !etween two o!"ects when there is no

motion. f s changes with the external force' Fext. f s ≤ s where

2 s > the coefficient of static friction 2 > the magnitue of the normal force

E%uality hols when the o!"ect is at the point of slipping. f smax@ > s

$%ample :

Consier a !lock on a rough surface. -pply an external force to the !lock.

2 if Fext f s max@ the o!"ect won5t mo$e.

2 as Fext increases' f s will increase until it reaches its maximum $alue.

2 When Fext > f s max@ the !lock will start to mo$e which is calle the point ofslipping.

2 Once the force starts to mo$e the force of friction is gi$en !y kinetic friction' f k .

2 8inetic Friction 8inetic Friction' f k is the force of friction !etween two o!"ects when there ismotion. f k > k N where

2 k > the coefficient of kinetic friction 2 N > the magnitue of the normal force

k is nearly inepenent of the $elocity of the o!"ect uner consieration.

f k is approximately constant for any gi$en pair of materials.

Note :

Dalues of s an k epen on the nature of the surfaces that are in contact. sually k s f k f k .

2 ru!!er on concrete s > 1.:' k > :.

2 waxe woo on wet snow s > :.10' k > :.1:

s an k are nearly inepenent of the area of contact !etween the two surfaces.


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3.3 inear momentum

and its

)rinciple o! Conser*ation

-pplication of ewton5s first an secon laws of motion in linear momentum 6inear momentum

)rinciple of conser$ation of linear momentum

-pplication of ewton5s First an ,econ 6aw5s of *otion

• From ewton5s ,econ 6aw+

Case1

• O!"ect at rest or in motion with constant $elocity !ut with changing mass.

• Example+ Rocket

( )

=

+

=

+

=

dt

dmv F

mdt

dm

v F

dt

vd m

dt

dmv F

:

Case

• O!"ect with constant mass !ut changing $elocity.

• Example+ Rocket

( )

am F

amv F

dt

vd m

dt

dmv F

=

+=

+

=

:

Case 3

• O!"ect at rest or in motion with constant $elocity an mass.

( )

dt

vd m

dt

dmv F

dt vmd F

dt

pd F

+=

=

=


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( ) ( )

:

::

=

+=

+

=

F

mv F

dt

vd m

dt

dmv F

• ewton5s First 6aw of *otion

malar p

dt

pd F

=

=

inear Momentum

;efinition +

*omentum > *ass x Delocity

#f ΣF > : p > mv > constant

*omentum is a $ector.

(he irection of the momentum is the same as the irection of the $elocity.

,.#. unit of momentum is kg m s =1 or s.

$%ample :

Fin the magnitue of the momentum of a cricket !all of mass 0: g thrown at : m s =1.

"olution :

7i$en m > :.0 kg v > : m s=1

*omentum > mass x $elocity p > mv

> :.0 x : s

> .0 s

$%ample :


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- 1.G kg !all was kicke with initial $elocity of 0: m s=1 at the angle of 3:° with the

horiHontal line. Calculate the initial momentum of the !all an also the horiHontal an

$ertical components of the initial momentum.

,olution+

$y $

7i$en m> 1.G kg$ > 0: ms=1 3::

:3:=θ $x

*omentum' p > m$

> 1.G x 0: kg ms=1

> /: kgms=1

IoriHontal component of the momentum + px > mvx

> mv kos θ

> 1.G 0:@ kos 3:°

> G1.J/ kg m s=1

or px > p kos θ

> /: kos 3:°

> G1.J/ s

Dertical component of the momentum +

py > mvy> mv sin θ

> 1.G 0:@ sin 3:°

> 3: kg m s=1

or py > p sin θ

> /: sin 3:°

> 3: s

)rinciple o! Conser*ation o!

inear Momentum


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,tates +

When the net external force on a system is Hero' the total momentum of that

system is constant.

Or

)ro$ie there are no external forces acting on a system' the total momentum !efore collisions e%uals the total momentum after collisions.

ewton5s First 6aw of *otion +

ΣF > : ∆ p > : p > constant

Expresse sym!olically +

Σ initial momentum > Σ final momentum

Σ pi > Σ p f

For a collision in$ol$ing two !oies +m1u1K m1u > m1$1 Km$

3.0 E6-,(#C L #E6-,(#C CO66#,#O

3.0.1 Collision

9 Collision is a process in which the colliing parties interact with each other$ery strongly an !riefly such that all the other forces can !e ignore in this process.

9 (he interaction force' e$en though !rief' epens strongly on time.

9 (he a$erage force of a collision process can !e calculate !y i$iing the

impulse' J !y the time inter$al of the collision' ∆t &

t

P

t

P P

t

J F

i f

∆

∆=

∆

−=

∆

=

9 #n a collision process !etween two !oies m1 an m' the external forces can !e

ignore' an as a result the linear momentum !efore an after the collision is conser$e.

m1u1 K mu > m1v1 K mv

where & u1 an u2 are $elocities of m1 an m !efore collision

v1 an v2 are $elocities of m1 an m after collision

9 On the other han' the kinetic energy may or may not !e conser$e in a

collision.

9 (wo types of collisions are elastic collision an inelastic

collision.

Example


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-n 1::=kg truck stoppe at a traffic light is struck from the rear !y a J::=kg car an the

two !ecome entangle. #f the smaller car was mo$ing at : mMs !efore the collision' what

is the spee of the entangle mass after the collision ?

,olution

Conser$ation of momentum

3.0. Elastic collision

9 -n elastic collision is that in which the momentum of the system as well as

kinetic energy of the system !efore an after collision is conser$e.

Conser$ation of momentum + m1u1 K mu > m1v1 K mv

Conser$ation of kinetic energy +N m1u1 K N mu

> N m1v1

K N mv

9 Elastic collision in one imension

[ ]

1

1

11

@111

/./

@J::1F::?

@:@?J::?

@?

:&?

−=

+=

+=

=+=

Σ=Σ

ms

mm

umv

uvmmum

P P f i


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Consier & m1 an m > masses of two non=rotating spheres

u1 an u > $elocities of m1 an m !efore collision

$1 an $ > $elocities of m1 an m after collision6et u1 is greater than u an are in the same irection+

*omentum of the system !efore collision > m1u1 K mu

*omentum of the system after collision > m1$1 K m$

-ccoring to the law of conser$ation of momentum+

m1u1 K mu > m1$1 K m$m1$1 2 m1u1 > mu 2 m$m1$1 2 u1@ > mu 2 $@ =======1@

,imilarly

8.E of the system !efore collision > N m1u1 K N mu

8.E of the system after collision > N m1$1 K N m$

,ince the collision is elastic' so the 8.E of the system !efore an after collision isconser$e.

(husN m1$1

K N m$ > N m1u1

K N mu

N m1$1 K m$

@ > N m1u1 K mu

@

m1$1=m1u1

> mu=m$

m1$1=u1

@ > mu=$

@

m1$1Ku1@ $1=u1@ > muK$@ u=$@ ======= @


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;i$iing e%uation @ !y e%uation 1@&

$1K u1 > u K $

From the a!o$e e%uation

$1 > u K $ =u1 3@

$ > $1 K u1 =u 0@

)utting the $alue of $ in e%uation 1@

m1 $1=u1@ > m u=$@

m1 $1=u1@ > mPu=$1Ku1=u@Q

m1$1=u1@ > mPu=$1=u1KuQm1$1=u1@ > m P u=$1=u1Q

m1$1=m1u1> mu=m$1=mu1 m1$1Km$1> m1u1=mu1Kmu

$1m1Km@ > m1=m@u1Kmu

( )( ) ( ) 1

1

1

11

1u

mm

m

u

mm

mm

v

+

+

+

−=

)v(um

)v )(uv(um

)u(vm

)u )(vu(vm

111

11111

−

−+=

−

−+


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9 Elastic collision in two imensions

Elastic collision in two imensions can !e analyHe !y using the fact that

momentum is a $ector %uantity.

Consier a glancing collision !etween two spheres of mass m1 an m &

$1 ' $ > $elocities !efore collision '

$51 ' $5 > $elocities after collision '

m is initially at rest' ∴ $ > :

the initial $elocity is along the x axis'

∴ the initial momentum along the y axis > :


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Conser$ation of momentum gi$es &

momentum alon+ % + m1$1x > m1$51x K m$5x

m1$1 > m1$51 cos θ1 K m$5 cos θ

momentum alon+ y + : > m1$51y K m$5y

m1$51 sin θ1 > m$5 sin θ

Conser$ation of energy gi$es &

inetic ener+y + N m1$1 > N m1$51

K N m$5

m1$1 > m1$51

K m$5

Example 1

- 3:::=kg truck mo$ing with a $elocity of 1: mMs hits a 1:::=kg parke car. (he impact

causes the 1:::=kg car to !e set in motion at 1G mMs. -ssuming that momentum is

conser$e uring the collision' etermine the $elocity of the truck after the collision.,olution

Example

- :: g tennis !all mo$ing with a spee of 1G mMs collies with a stationary !all of :: g

in an elastic collision. (he tennis !all is scattere at an angle of 0G o from its originalirection with the spee of

G mMs. Fin the final spee magnitue an irection@ of the struck !all.

,olution


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Conser$ation of momentum along x +

m1$1 > m1$51 cos θ1 K m$5 cos θ

:.1G@ > :.G cos 0Go

@K :. $5 cos θ @

$5 cos θ > 3 = :.G cos 0Go @

:.

$5 cos θ > .// i@

Conser$ation of momentum along y +

m1$51 sin θ1 > m$5 sin θ

:.G sin 0Go @ > :. $5 sin θ

$5 sin θ > :.G sin 0Go @

:. $5 sin θ > :.0 ii@


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3.0.3 #nelastic collision

9 -n inelastic collision is that in which the momentum of the system

!efore an after collision is conser$e !ut the kinetic energy !efore an after collision is

not conser$e.

Conser$ation of momentum'

m1u1 K mu > m1K m@ $

8inetic energy is not conser$e'

Σ 8.E !efore collision ≠ Σ 8.E after collision

N m1u1 K N mu

≠ N m1 Km@v2

some of the kinetic energy is transforme into other forms of energy suchas heat or soun@

9 #nelastic collision in one imension

#n a completely inelastic collision the two o!"ects stuck together after collision.

-ccoring to the law of conser$ation of momentum'

m1u1 K mu > m1 K m @ $

#f m is at rest' u > : &


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8inetic energy !efore collision ' 8 i > N m1 u1

8inetic energy after collision' 8 f > N m1K m@ v

9 Aallistic penulum Aallistic penulum is a e$ice in$ente !y Aen"amin Ro!ins in 10 to measure the

spee of a !ullet.


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(he !lock is at rest' so its $elocity is Hero.

Conser$ation of momentum&

m$i > * K m@ $f i@

8inetic energy of !oth !ullet an the !lock after collision&

8.E > N M K m@ $f

Conser$ation of energy&

8inetic energy> 7ra$itational potential energy

N M K m@ $f > * K m @g h

$f > √ g h ii@

Example 1

#n a !allistic experiment' suppose that &

h > G.:: cm'm > G.:: g

* > 1.:: kg. Fin the a@ initial spee of the !ullet' vi !@ the loss in energy ue to the collision g > J.1 mMs S

,olution

!@ (he loss in energy ue to the collision' ∆8

∆8 > 8 i 2 8 f


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> N m$i = N m K *@ $f

> N m$i = N m K *@ gh@

> N PGx1:=3@1JJ@ =Gx1:=3 K 1.::@ @J.1@Gx1:=@ > JJ.: = :.G

> J.G B

Example

7ranny m>: kg@ whiHHes aroun the rink with a $elocity of / mMs. ,he suenly

collies with -hma m>0: kg@ who is at rest irectly in her path. Rather than knock himo$er' she picks him up an continues in motion without T!raking.T ;etermine the $elocity

of 7ranny an -hma. -ssume that no external forces act on the system so that it is an

isolate system.

,olution

Example 3

(wo cars approaching each other along streets that meet at a right angle collie at theintersection. -fter the crash' they stick together. #f one car has a mass of 10G: kg an an

initial spee of 11.G mMs an the other has a mass of 1G: kg an an initial spee of 1G.G

mMs' what will !e their spee an irection immeiately after impact ?,olution

(he x component of the $ector&


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m1$1 > m1K m@ *x

(he y component of the $ector'

m1v1 > m1K m@ vy

3.0.0 Coefficient of Restitution


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9 Coefficient of restitution is the ratio of the ifferences in $elocities !efore an after the

collision

9 (he coefficient of restitution will always !e !etween Hero an one'

: ek 1 @

9 - perfectly elastic collision has a coefficient of restitution of 1'

ek > 1

9 - perfectly inelastic collision has a coefficient of restitution of :'

ek > :

3.G #mpulse(he impulse of the force F e%uals the change in the momentum of the particles

When a !ase!all hits a !at

• or when two !illiar !alls collie ' they exert forces on each other o$er a $ery

short time inter$al. Forces of this type ' which exist only o$er short time ' are often calle

impulsi$e forces.

• -ccoring ewton5s secon law

F > U m$@ > Up

Ut Ut

where F is the net force applie to an o!"ect an Up is its change

in momentum uring a time Ut

F Ut > Up

(he %uantity F Ut is calle the impulse . #t is the prouct of the force

an the time inter$al Ut o$er which the force acts.

7raph of force $s time


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• -$erage an instantaneous force uring a typical !rief collision !etween two

mo$ing !oies.

(he area uner the cur$e of force $ersus time is e%ual to the impulse.

• since the area of the rectangle whose height is the a$erage force e%uals the area

uner the cur$e ' we can replace the instantaneous force !y the a$erage force to o!tain

the impulse.

Documents

Chapter 3 Force Impulse Momentum