General Physics 1

• C3 – Interactions transfer momentum

• C4 - Particles and Systems

General Physics 2

Solutions to HW

• Download ProbViewer 1.4• www.physics.pomona.edu/sixideas/sicpr.html

• Password for daily HW problems• kLLxJkS8

General Physics 3

Displacement

!

"r r =

r r 2#

r r 1

"x

"y

"z

$

%

& & &

'

(

) ) )

=

x2

y2

z2

$

%

& & &

'

(

) ) )

#

x1

y1

z1

$

%

& & &

'

(

) ) )

=

x2# x

1

y2# y

1

z2# z

1

$

%

& & &

'

(

) ) )

(x2, y2, z2)

(x1, y1, z1)

General Physics 4

Velocity

• Change in position in a small timeinterval

• Column vector

!!!

"

#

$$$

%

&

=

!!!

"

#

$$$

%

&

=

!!!

"

#

$$$

%

&

==

dtdz

dtdy

dtdx

dz

dy

dx

dtv

v

v

dt

rdv

z

y

x1

rr

General Physics 5

Momentum

• “Quantity of motion”• Vector• Defined as

• in terms of mass and velocity• Units are kg·m/s

!

r p " m

r v

General Physics 6

Momentum - skill building problem

• What is the momentum of an 8-kgbowling ball rolling at 2 m/s?

General Physics 7

Think-Pair-Share

• What is the magnitude of themomentum of a 28-g sparrow flyingwith a speed of 8.4 m/s?

General Physics 8

Impulse

• Amount of momentum that aninteraction (A) transfers to eitherparticle during a short interval

!

dr p [ ]

A

General Physics 9

Change in momentum

• Multiple interactionscontribute their ownimpulses resulting ina change inmomentum

!

dr p = d

r p [ ]

A+ d

r p [ ]

B+ d

r p [ ]

C+ ...

General Physics 10

Impulse - skill building problem

• What is the impulse on an 8-kg ballrolling at 2 m/s when it bumps into apillow and stops?

General Physics 11

Impulse - skill building problem

• How much impulse stops a 50-kg cartonsliding at 4 m/s when it meets a roughsurface?

General Physics 12

Impulse Questions

• In terms of impulse and momentum,why do airbags in cars reduce thechances of injury in accidents?

• When jumping from a significant height,why is it advantageous to land with youknees bent?

General Physics 13

Financial Analogy

• Particle’s momentum is a person’s netfinancial worth

• Interaction is a financial interaction• Increases one’s net worth at the expense

of the other’s• Inpulse is a check that is written or

received• Momentum in general is like money

General Physics 14

Force

• Rate at which momentum flows

!

r F A =

dr p [ ]

A

dt

General Physics 15

Impulse

• Change in momentum• Vector quantity• The magnitude of the impulse is equal to the area

under the force-time curve• The impulse can also be found by using the time

averaged force

• This would give the same impulse as the time-varying force does

[ ] dtFpd AA!=

rr

General Physics 16

Think-Pair-Share

• An estimated force-time curve for a baseball struckby a bat is shown. From this curve, determine theimpulse delivered to the ball.

General Physics 17

Mass and Weight

• In physics, mass and weight arecompletely distinct concepts

• Where mass is a scalar, is thegravitational field vector

• is the weight acting on an object

!

r F g = m

r g

!

r g

!

r F g

General Physics 18

Momentum flow and motion - 1

• Multitap

General Physics 19

Momentum flow and motion - 2

• Reservoir

General Physics 20

Multitap and Reservoir - Exercise C3X.7

• Imagine a pool ball that isinitially rolling due northon a pool table. Youropponent blows constantlydirectly eastward on theball. Model the ball as aparticle moving initiallynorthward but experiencesa constant eastward force.Sketch momentum arrowsfor the ball at successivetimes using the multitapand reservoir models.

General Physics 21

Solution

General Physics 22

Conservation of Momentum

• A system’s total momentum is the vector sumof the momenta of its constituent particles

• The law of conservation of momentum:• The total momentum of an isolated system is

conserved; that is, it does not change in time.

!

r p tot "

r p 1

+r p 2

+r p 3

+ ...+r p N

General Physics 23

Conservation of Momentum

• The total momentum of a system is constant.• For collisions, use

• In component form, the total momenta in eachdirection are independently conserved

ffii

afterbefore

pppp

pp

2121

rrrr

rr

+=+

=

fzizfyiyfxix pppppprrrrrr

===

General Physics 24

Think-Pair-Share

• A 95-kg halfback moving at 4.1 m/s onan apparent breakaway for atouchdown is tackled from behind.When he was tackled by an 85-kgcornerback running at 5.5 m/s in thesame direction, what was their mutualspeed immediately after the tackle?

General Physics 25

The Center of Mass

• There is a special point in a system orobject, called the center of mass, thatmoves as if all of the mass of thesystem is concentrated at that point

• The system will move as if an externalforce were applied to a single particle ofmass M located at the center of mass• M is the total mass of the system

General Physics 26

Motion of the Center of Mass

• A projectile is fired into theair and suddenly explodes

• With no explosion, theprojectile would follow thedotted line

• After the explosion, thecenter of mass of thefragments still follows thedotted line, the sameparabolic path the projectilewould have followed withno explosion

General Physics 27

Center of Mass

• is the system’s center of mass

• are the positions of the system’s Nparticles

• are the masses• is the system’s mass

!

r r CM

"1

Mm1

r r 1+ m

2

r r 2

+ ...+ mN

r r N( )

!

r r CM

!

r r 1,r r 2,...,

r r N

!

m1,m

2,...,m

N

!

M " m1+ m

2+ ...+ m

N

General Physics 28

Center of mass - column vectornotation

!

xCM

yCM

zCM

"

#

$ $ $

%

&

' ' '

=1

Mm1

x1

y1

z1

"

#

$ $ $

%

&

' ' '

+ m2

x2

y2

z2

"

#

$ $ $

%

&

' ' '

+ ...+ mN

xN

yN

zN

"

#

$ $ $

%

&

' ' '

(

)

* * *

+

,

- - -

!

r r CM

"1

Mm1

r r 1

+ m2

r r 2

+ ...+ mN

r r N( )

General Physics 29

Center of Mass, Example

• Both masses are onthe x-axis

• The center of massis on the x-axis

• The center of massis closer to theparticle with thelarger mass

General Physics 30

Center of Mass

• Dick Fosbury introduced a new highjump technique - the Fosbury Flop - atthe 1968 Olympics. Raised worldrecord 30 cm!

General Physics 31

Center of mass example

• Insert example on p. 67 to introduceactivity

General Physics 32

Center of mass activity - 1

• You will need• Meter stick• Fulcrum• Mass hangers• masses

General Physics 33

Center of mass activity - 2

• Use column vector notation for twoexamples• Same mass• Different masses

• Find location of and compare toexperimental result

!

r r

General Physics 34

System’s Momentum

• A system’s center of mass responds toits external interactions exactly as apoint particle would respond to thoseinteractions

!

r p tot = M

r v CM

General Physics 35

Interacting with the earth-1

• Does the earth’s momentum changewhen interacting with you?• It must if momentum is conserved!• But if I jump, the earth doesn’t move!

• Use a reference frame that includes youAND the earth that is fixed to the CM ofyou and the earth• Freely floating frame

General Physics 36

Interacting with the earth - 2

• Assume• Person’s mass ~ 50 kg• Person jumping ~ 3 m/s vertically• Person’s impulse upward = 150 kg m/s

• Earth’s impulse is the same, 150 kg m/s• Calculate velocity

• Impossible to detect

( ) ( )sm

kg

smkg

m

pmagvmagv

earth

earthff

24

241025

106

150 !"#

"

$==%

rr