experimental physics ep1 mechanics - system of particles - - conservation of momentum ·...

Experimental Physics - Mechanics - System of Particles - Conservation of Momentum 1

Experimental Physics EP1 MECHANICS

- System of Particles -- Conservation of Momentum -

https://bloch.physgeo.uni-leipzig.de/amr/


Center of mass

m mm mm m

3m 3m

4x1x 2x 3x 5x 6x 7x å å=i i

iiicm xmmX

471=

+×+×

=mmmmXcm

44

7621=

+++=cmX

233211 =

++=cmX 6

37652 =

++=cmX 4

6)62(3, =

+==

åå

mm

MXM

Xi

icmicm

m1 m3m2

21 mm +å å=i i

iiicm rmmR !!


3mm

Center of mass

å å === cmiiii MghhmgghmU

grF!

å= iighmU0

å D+= ii hgmUU 0q

qq sin20 mgUU -=

0cos2 =-= qq

mgddU

2m


R

x

y

Continuous bodiesdm

ò

ò

ò

ò== L

L

L

L

cm

rdr

rdrr

dm

dmrR

0

0

0

0

)(

)(

!!

!!!!!

r

r0 L

221

0

20

0 0

0 0 LLL

dx

xdxX L

L

cm ===òò

rr

r

r

6)(

)( R

xdm

xxdmX R

R

R

Rcm -==

òò-

-

62/)(0 R

mMRmMXcm -=

-×-+×

=

Negative mass:


Motion of the center of mass

cmR!

extF!

å=i

iicm rmMR !!

å=i

ii

cm

dtrdm

dtRdM

!!

å=i

ii

cm

dtrdm

dtRdM 2

2

2

2 !!

( )åå +==i

iii

icm FFFAM ext,,int

!!!!

extA!

cmAMF!!

=net,ext

Under influence of an external force thecenter of mass of a system moves like a pointparticle with a mass equal to the total mass ofthe system and irrespective of internal forcesacting within the system.


Conservation of momentum

vmp !!=

1v!

net)( Fam

dtvmd

dtpd !!

!!===

Linear momentum:

12F! 21F!

2112 FF!!

-=

dtpd

dtvmdF 22

12)( !!!==

constppdtpd

dtpd

=+Þ-= 2112 !!!!

å å === cmiii VMvmpP!!!!

net,extFdtVdM

dtPd cm

!!!

==

The law of conservation of momentum:

constVMvmP cmii ===å!"!

2v!

interaction


0 5 10 15 20 25 30 35 40 45

-8

-6

-4

-2

0

2

4

6

8 tv!

0v!

y

x

q0

Projectile with explosion

22

0000 )cos(2)(tan x

vgxyyq

q -+=

x

toptop v

xxgyy

2

2

2 2)( -

-=

m2

m

m

2102 vmvmvm !!!+=

xx mvmv 202 =

xx vv 02 2=

?


0 5 10 15 20 25 30 35 40 45

-8

-6

-4

-2

0

2

4

6

8 tv!

0v!

y

x

q0

Projectile with explosion

22

0000 )cos(2)(tan x

vgxyyq

q -+=

x

toptop v

xxgyy

2

2

2 2)( -

-=

m2

m

m

2102 vmvmvm !!!+=

xx mvmv 202 =

xx vv 02 2=

Experimental Physics - Mechanics - Elastic and Inelastic Collisions 9

Rocket propulsionv

0M

gu

constvmP ii ==å !!

ug is relative to the rocket;

dtdmR =

At an arbitrary time t

( ) ( )( )dvvdmmdmuvmv g +-+-=

At a time t+dt

gg Rudtdmu

dtdvm == - the thrust

ò -=-=t

RtMdmMm0 00

( )òò --=t

g

v

vdtRtMRudvf

i 0

10

extF!

+

÷÷ø

öççè

æ-

+=RtM

Muvv gif0

0ln

0 5 10 15 20 25 30 35 40 45 50 55

0

10

20

30

40

spee

d, m

/s

time, s

R is the rate at whichgas is exhausted


m1

Skaters on ice

m2

m1 m21v!

2v!

0=å ivm!

2211 vmvm =

tx

mtx

m 22

11 =

extF!

cmcmext VAmF Þ=!!

( ) ( ) cmcmcm VmmVvmVvm )( 212211 +-=-+--

cmVvu!!!

+= 02211 =+- vmvm

( ) åååå å ++=+== iicmcmiiicmiiiik vmVVmvmVvmumE 2222

21

21

21

21

rel,2

21

kcmk EMVE += Ek,rel is reference-independent


Ø The center of mass of a system moves like a point-like

object having the total system mass.

Ø If there is no net force acting upon the system, then

the linear momentum is conserved.

Ø The total kinetic energy of the system can be written

as the sum of the kinetic energy associated

with the center of mass plus the kinetic

energy associated with the particle

movements in the center-of-mass

reference system.

To remember!


Elastic collision

constP

constE

total

totalk

=

=,

))(())(( 1111122222 fifiifif vvvvmvvvvm +-=+-

2222

12112

12222

12112

1iiff vmvmvmvm +=+constE totalk =,

constPtotal = iiff vmvmvmvm 22112211 +=+

)()( 111222 fiif vvmvvm -=-

Þ+=+ fiif vvvv 1122 )( 1212 iiff vvvv --=-

Relative speed of approach is equal to relative speed of recession.


Center-of-mass reference frame

constVMvmP cmii ===å!"!

icm uV 1+ icm uV 2+

ii pp 21 -=

÷÷ø

öççè

æ+=+=+=

21

21

2

22

1

21

222

211

, 21

21

2222 mmp

mp

mpvmvmE i

iiiiik

÷÷ø

öççè

æ+=+=+=

21

21

2

22

1

21

222

211

, 21

21

2222 mmp

mp

mpvmvm

E fffff

fk

No external forces!

Elastic collision Þ Þ21

21 fi pp =

ïî

ïíì

±=

±=

if

if

pp

pp

22

11

ïî

ïíì

±=

±=

if

if

uu

uu

22

11

In CoM reference frame the velocities of each object are simply reversed by collision.

Experimental Physics - Mechanics - Selected problems 14

Astroblaster

0v

0v0v

0v

2m

1m2v

1v

22110201 vmvmvmvm +=-

)( 1212 iiff vvvv --=-

00012 2)( vvvvv =---=-

02v

0v3v3m

2m

332203022 vmvmvmvm +=-

00023 3)2( vvvvv =---=-

021

211

3 vmmmmv

+-

=021

21

2;03

vvvmm

===

32

3202

22mmmmvv

+-

=

032

32

3;02

vvvmm==

=

( ) 920303 == vvhh


Perfectly inelastic collision

iv1 iv2 cmV

cmii MVvm =å

2

22

1

21

222

211

, 2222 mp

mpvmvmE iiii

ik +=+=)(22 21

2221

, mmpVmmE cmfk +

=+

=

02 =iv2)(2

2 211

21

1

21

,1,

iikfk

vmmm

mmmEm

E+

=+

=21

21

mmvv ii

=-= 0, =fkE

bv hghmmvm

mmmE i

fk )(2 21

211

21

1, +=

+=


Coefficient of restitution

)( 1212 iiff vvevv -×-=-e = 1 – elastic collisione = 0 – perfectly inelastic collision

1h2h1v

2v

120 evvvplate =Þ=

2

1

2

1

1

2

hh

mghmgh

vve ===

Table tennisStainless tablee = 0.92

Bat-bat COR e < 0.5

?)( -=D efEk MpE ik

2

, = MpeE fk

22

, = )1( 2, -=D eEE ikk

ò=f

i

t

tdtFI!!

- impulse ò -==f

i

t

t if ppdtdtpdI !!!!


Inelastic collision in 2D

cmii VMvmP!!!

==å

1p!

2p!

°= 30q km/h401 =v

km/h70tan

12 ==

qvv

P!


Elastic collisions in 2D

2q

m2

m1iv1!

m1

02 =iv!

fv1!

fv2!

1q

constvmP ii ==å !!

îíì

+==

+==

2,221,11

2,221,11,11

sinsin0

coscos

qq

qq

fxfxy

fxfxixx

vmvmP

vmvmvmP1

2

3 2222

12112

12112

1ffi vmvmvm +=

21 mm = 22

21

21 ffi vvv +=ffi vvv 211

!!!+=

iv1!

fv2!

fv1!


Ø During inelastic collisions the total kinetic energy is not

conserved. It is conserved during elastic collision.

Ø In a perfectly inelastic collision the bodies stick together and

move with the velocity of COM.

Ø In the COM reference frame, upon 1D elastic collision

the bodies change their velocities to

the opposite ones.

Ø For non-central collisions one has to

consider conservation of all components

of the linear momentum.

To remember!

experimental physics ep1 mechanics - system of particles - - conservation of momentum ·...

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