Chap 15 Kinetics of a particle impulse and momentum

15-1. principle of linear impulse and momentum15-1. principle of linear impulse and momentum(1) Linear impulse I ,N S(1) Linear impulse I ,N S

The integral I =∫F dt is defined as the linear impulse which measure the The integral I =∫F dt is defined as the linear impulse which measure the effect of a force during the time the force acts .effect of a force during the time the force acts .

2

1)(

t

tdttFI

(2) Linear momentum L , kg/s

The form L=m v is defined as the linear momentum of the particle .

(3) Principle of linear impulse and momentum the initial momentum of the particle at time T1 plus the vector sum of all the time integral t1 to t2

is equivalent to the linear momentum of the particle at time t2 .

2

1 21

t

tmvdtFmv

Equation of motion for a particle of mass m is ∑F =m a = m dv /dt ∑F dt = m dv

Integrating both sides to yield .

2

1

2

1 12

t

t

v

vmvmvvmddtF

2

1 21

t

tmvdtFmv

In x,y,z, components

2

2

11

2

2

11

2

2

11

)()(

)()(

)()(

z

t

t zz

y

t

t yy

t

t xxx

vmdtFvm

vmdtFvm

vmdtFvm

15.2 Principle of linear impulse and momentum for a system of particles.

Equation of motion for a system of particle is

dtivd

imiaimiF

12

)(

)(

2

1

)(

)(

2

1

2

1

iiii

v

v

t

t

v

v iiiii

vmvm

vmvdmdtFi

i

i

i

ivdimdtiF

or

1...........21

2

1 ivimivim

t

t idtF

The location of the mass G of the system is

Substitute Eq (2) to (1) to get

irimrmG

imm

2............or ivimvi

dt

drim

dt

rdm

GG

2

1 21

ttG G

vmdtFvm

15.3 Conservation of linear momentum for a system

1. Internal impulses

The impulse occur in equal but opposite collinear pairs.

2

1

2

1

2

12112

tt dtfdt

tt f

tt dtF

2. Non impulsive forces The force cause negligible impulses during the very short time

period of the motion studied . (1)weight of a body. (2)force imparted by slightly de formed spring having a relatively small stiffness ( Fs = ks ). (3)Any force that is very small compared to other longer impulsive forces.3.Impulsive forces The forces are very large , act for a very short period of time and

produce a significant charge in momentum. Note :when making the distinction between impulsive and nonimpulsive forces it is important to realize that it only applied during a specific time. Conservation of linear moment. The vector sum of the linear moments for a system of particles remain constant throughout the time period t1 to t2

15.4 Impact

Impact

(1) Definition. Two bodies collide with each other during a very

short internal of time which causes relatively large impulsive

forces exerted between the bodies.

(2) Types.

(A). Central impact. The direction of motion of the mass

center of the two colliding particles is along the line of

impact.

(B). Oblique impact. The motion of one or both the particles

is at angle with the line of impact.

Central Impact

To illustrate the method for analyzing the mechanics of impact , consider

the case involving the central of 2 smooth particles A and B.

conservation of momentum of the system of panicles A and B

mAvA1 mAvA2

Require

A B

vA1>vA2

A B

∫Pdt -∫Pdt

Effect of A on B Effect of B on A

V

A B

Before impact

Deformation impulseMaximum deformation

2211 BBAAbbAA vmvmvmvm

Principle of impulse and momentum for panicle A (or B)

(a) Deformation please for A

(b) Restitution place for A

vmPdtvm

Pdt

AAA

2AmvRdtmv

deformation impulse is

Restitution impulse is

Define

Similarly by considering particles B We have

vmvmPdt AAA

2AAA vmvmRdt

vv

vve

mvvm

vmvm

Pdt

Rdt

A

A

AA

AAA

1

2

1

2

vvevv

vevev

vvevv

vv

vve

BB

AA

AA

A

A

21

21

21

1

2

)(

)1(.1

)(

=relative velocity just after impact

=relative velocity just before impact

In general

Elastic impact

11

22

1221

12)1.(2

BA

AB

BBAA

BB

vv

vve

evvvev

evvve

1)1(

10

e

e

RdtPdt

Plastic impact

3. Oblique impact

Plastic Impact (e = 0):The impact is said to be inelastic or plastic when e = 0. In this case there is no restitution impulse given to the particles

0)1( e

0Pdt

(∫Rdt = 0), so that after collision both particles couple or stick together & move with a common velocity.

Oblique Impact. When oblique impact occur between2 smooth particles, the particle move away form each other with velocities having unknown direction as well As unknown magnitudes. Provided the initial velocities

are known,4 unknown are present in the problem.

Line of impact

vB2vA2

vA1 vB1

Plane of contact

+ =

mAvAx1

mAvAy1

∫Fdt

mAvAy2

mAvAx1

+ =∫Fdt

mBvBx2

mBvBy2

mBvBy1

mBvBx1

15-5 angular momentum

1. angular momentum

Moment of the particle’s linear momentum about point 0 or other point.

2. scalar formulation

Assume that the path of motion of the particle lies in x y-plane.

Magnitude of angular momentum Ho is

Ho= Ho = d(mv)

Angular momentum=moment of momentum

Ho=r x mv= i j k

rx ry rz

mvx mvy mvz

15-6 Moment of a force and Angular momentum

Equation of motion of a particle is ( m=constant )

By performing a cross-product multiplication of each side of this

equation by the position vector r , We have the moments of forces

About point O

VmF

F

)( VmxrFr

oM

)( vmxrHo

)( vxmr

dt

dHo

vxmrvxmr vxmrvxrm )(

oo MvxmrH

oo HMor

Resultant moment about pt.o of all force

Time rate change of angular momentum of the particle about pt.o

Recall

LF