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Kinetics of a Particle: Impulse and Momentum

Linear momentum

Linear impulse

mL v

dt I F

( )d dm mdt dt

LF a v

2

12 1 2 1

t

tdt m m F L L v v

Newton’s 2nd law: The resultant of all forces acting on a particle is equal to its time rate of change of linear momentum.

Chungnam National University

Principle of Linear Impulse and momentum

2

11 2

t

tm dt m v F v

2

1

2

1

2

1

1 2

1 2

1 2

( ) ( )

( ) ( )

( ) ( )

t

x x xt

t

y y yt

t

z z zt

m v F dt m v

m v F dt m v

m v F dt m v

2

12 1 2 1

t

tdt m m F L L v v

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Linear Impulse and momentum: Examples

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Principle of Linear Impulse and momentum for a system of particles

ii i i

dmdt

vF f

2

11 2( ) ( )

t

G i Gtm dt m v F v

( )ii i

d mdt

vF f for particle i

For system of particles

2

121

( ) ( )t

i i i i itm dt m v F v

G i i

G i i

m m

m m

r r

v vim mwhere

For rigid body

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Conservation of Linear Momentum for a system of particles

21( ) ( )i i i im m v v

1 2( ) ( )G Gv v

If the resultant force on a particle is zero during an interval of time, 2

121

( ) ( )t

i i i i itm dt m v F v

G i im mv vFor rigid body, since

When is the resultant force on a particle zero during an interval of time? (1) Particles collide or interact.

(2) External impulse is negligible, when the time period is very short2

11 2

t

A A A Atm dt m v F v

2

11 2

t

B B B Btm dt m v F v

1 1 2 2A A B B A A B Bm m m m v v v v

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Conservation of Linear Momentum: example

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Conservation of Linear Momentum: example

1 1 ( )proj proj Block Block proj Block proj Blockm m m m m m v v v v v

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Impact

Definition of impact: collision between two bodies is characterized by the generation of relatively large contact forces that act over a very shot interval of time.

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Central Impact

1 1 2 2( ) ( ) ( ) ( )A A B B A A B Bm v m v m v m v

Law of conservation of linear momentum is valid. Why?Internal impulse of deformation and restitution cancel, during collision

How many unknowns in the above equation?

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Central Impact

1( )A A Am dt m v P v

Deformation PeriodFor particle A

For particle B

1( )B B Bm dt m v P v

1( )A A Adt m m P v v

1( )B B Bdt m m P v v

-

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Central Impact

2( )A A Am dt m v R v

Restitution Period

For particle A

For particle B

2( )A A Adt m m R v v

2( )B B Bm dt m v R v

2( )B B Bdt m m R v v

-

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Coefficient of Restitution

2

1

( )( )

A

A

Rdt v vev vPdt

2

1

( )( )

B

B

Rdt v vev vPdt

2 2

1 1

( ) ( )( ) ( )

B A

A B

v vev v

Coefficient of restitution: the ratio of the restitution impulse to the deformation impulse.

For particle A

For particle B

(1)

(2)

Eliminate v using eqs. 1 & 2

relative velocity of separationerelative velocity of approach

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Central Impact

1 1 2 2( ) ( ) ( ) ( )A A B B A A B Bm v m v m v m v

e=1: restitution impulse = deformation impulse

No energy loss – perfectly elastic

e=0: plastic impact 100% energy loss 2 2( ) ( )A Bv v v

Summary of Central impact problem

2 2

1 1

( ) ( )( ) ( )

B A

A B

v vev v

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Impact example

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Impact example

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Oblique ImpactX direction

1 1 1( ) ( ) cosAx Av v 1 1 1( ) ( ) cosBx Bv v

Y direction1 1 1( ) ( ) sinAy Av v

1 1 1( ) ( ) sinBy Bv v

2 2 2( ) ( ) cosAx Av v

2 2 2( ) ( ) cosBx Bv v

2 2 2( ) ( ) sinAy Av v

2 2 2( ) ( ) sinBy Bv v unknowns

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Oblique ImpactX direction

1 1 2 2( ) ( ) ( ) ( )A Ax B Bx A Ax B Bxm v m v m v m v

2 2

1 1

( ) ( )( ) ( )

Bx Ax

Ax Bx

v vev v

Y direction

1 2( ) ( )A Ay A Aym v m v

1 2( ) ( )B By B Bym v m v

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Angular Momentum

( ) ( )( )oH d mv

The moment of the linear momentum L about O is defined as the angular momentum Ho of particle P about O.

o

o x y z

x y z

m

i j kr r r

mv mv mv

H r v

H

Unit: kg(m/s)m=kg(m/s2)ms=Nms

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Relation Between Moment of a Force and Angular Momentum

( )

o

o

m

md m m mdt

F v

M r F r v

H r v r v r v

ooM H F L

The resultant moment about the fixed point O of all forces acting on a particle is equal to its time rate of change of angular momentum of the particle about O.

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System of Particles

( ) ( ) ( )i i i i o ir F r f H

( ) ( ) ( )i i i i i o r F r f H

For the particle i

ooM H

For system of particles

The sum of the moments about point O of all the external forces acting on a system of particles is equal to the time rate of change of the total angular momentum of the system about point O.

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Angular Impulse and Momentum Principles

2

1

2

1

2 1

1 2

( ) ( )

( ) ( )

t

o o ot

t

o o ot

dt

dt

M H H

H M H

Principle of angular impulse and momentum

ooo

ddt

HM H oodt dM H

2 2

1 1

( )t t

ot tangular impulse dt dt M r F

2

11 2( ) ( )

t

o o otdt H M H

For a particle

For system of particles

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2

1

2

1

1 2

1 2( ) ( )

t

t

t

o o ot

m dt m

dt

v F v

H M H

2

1

2

1

2

1

1 2

1 2

1 2

( ) ( )

( ) ( )

( ) ( )

t

x x xt

t

y y yt

t

o o ot

m v F dt m v

m v F dt m v

H M dt H

Vector formulation

Scalar formulation (2D case)

Impulse and Momentum Principles

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Conservation of Angular Momentum

1 2( ) ( )o oH H

When the angular impulses acting on a particle are all zero during the time t1 to t2, then

1 2( ) ( )o o H H

From t1 to t2, the particles angular momentum remains constant.

Conservation of angular momentum of a system of particles

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Conservation of Angular Momentum: example

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Conservation of Angular Momentum: example