2008 Physics 2111 Fundamentals of Physics Chapter 9 1 Fundamentals of Physics Chapter 9a Systems of...

2008 Physics 2111 Fundamentals of Physics

Chapter 9 1

Fundamentals of Physics

Chapter 9a Systems of Particles

1. A Special Point

2. The Center of Mass

3. Newton’s 2nd Law for a System of Particles

4. Linear Momentum

5. Linear Momentum of a System of Particles

6. Conservation of Linear Momentum

7. A Rocket

8. External Forces & Internal Energy Changes

Review & Summary

Questions

Exercises & Problems


Chapter 9 2

The Center of Mass

• The center of mass of a body or system of bodies is the point that moves as though all of the mass were concentrated there and all external forces were applied there.


Chapter 9 3

The Center of Mass

Consider 2 particles separated by distance d:

The center of mass, xcom, of the system is the point that moves as though all of the mass were concentrated there and all external forces were applied there.

xm x m x

m mcom

1 1 2 2

1 2


Chapter 9 4

Center of Mass (cm)

comx

comx

xcom is closer to the larger mass.

xm

m mdcom

2

1 2


Chapter 9 5

The Center of Mass

Many particles distributed along the x-axis:

The center of mass, xcom, of the system is the point that moves as though all of the mass were concentrated there and all external forces were applied there.

Many particles distributed in 3 dimensions:

i

n

iicom xm

Mx

1

1

i

n

iicom rm

Mr

1

1

i

n

iicom

i

n

iicom

i

n

iicom

zmM

z

ymM

y

xmM

x

1

1

1

1

1

1


Chapter 9 6

Center of Mass?

20142803

16241803

15483

ii

ii

i

ym

xm

mM

1.115

16

i

iicom m

xmx 3.1

15

20

i

iicom m

ymy


Chapter 9 7

CM of Solid Bodies

Uniform density case:

dmzM

z

dmyM

y

dmxM

x

com

com

com

1

1

1

dmrM

rcom 1

dVV

MdVdm


Chapter 9 8

Newton’s 2nd Law for a System of Particles

r M m r

M r m r

M v m v

M a m a F F

com ii

n

i

com ii

n

i

com ii

n

i

com ii

n

i ii

n

net

11

1

1

1 1

comnet aMF

Net Force of all external forces

Internal forces cancel (Newton’s 3rd Law)

Total mass of the system

Acceleration of the center of mass


Chapter 9 9

Newton’s 2nd Law for a System of Particles

The center of mass of a system moves like a particle of mass M under the influence of the net external force acting on the system.

• Net of all external forces

Internal forces cancel (Newton’s 3rd Law)

• Total mass of the system

• Acceleration of the center of mass

comnet aMF


Chapter 9 10

Linear Momentum

Linear momentum of a particle:

The net force acting on a particle equals the time rate of change of the particle’s linear momentum:

dpdt

ddt

dvdtm v m ma

Fnet

dpdt

p mv

Newton’s 2nd Law


Chapter 9 11

Linear Momentum of a System of Particles

The linear momentum of a system of particles equals the product of the total mass and the velocity of the center of mass:

Total linear momentum of a system of particles:

P p m vi

i

n

ii

n

i

1 1

But M v m vcom ii

n

i

1

P M vcom

Differentiating: com

comcom aM

dt

vdMvM

dt

d

dt

Pd

netFdt

Pd


Chapter 9 12

Conservation of Momentum

Law of conservation of momentum: If no net external force acts on a system of particles, the total linear momentum of the system cannot change:

If the component of the net external force on a closed system is zero along an axis, then the component of linear momentum of the system along that axis cannot change.

netFdt

Pd

constantP

if PP


Chapter 9 13

A shell explodes at its highest point.

M2

M

2M

the com continues on its way!

Coordinates of the explosion:

Momentum in the x-direction at the explosion:

0

20

21

21 2sin

g

vRxE

g

vyE 2

sin 200

000 2cos V

MvM

?maxx

g

yVxtVxx E

EE

200max Projectile motion from the explosion:

v0 = 20 m/s

= 600


Chapter 9 14

Method 2: the com continues on its way!


Chapter 9 15

The com continues on its way!


Chapter 9 16

no friction “the system” = person + boat

When the person walks to the front of the boat, how far is the boat from the pier?

changenot does com theofmotion 0 netF

0restat initially is system the comv

afterxbeforex comcom

12060

312060

12060

5.31205.660

dd

md 5.2

?d


Chapter 9 17

Conservation of ??

g

WM

g

wm

Initially both are moving with velocity v0; then the man starts to run with velocity vrel with respect to the car.

momentum before: 0vmM

momentum after: relvvmvM mM

vmvv rel

0


Chapter 9 18

ROCKETS


Chapter 9 19

Rockets

Further investigation and experimentation have confirmed the findings of Isaac Newton in the 17th century and it is now definitely established that a rocket can function in a vacuum as well as in an atmosphere. The Times regrets the error

• for after the rocket quits our air and really starts on its longer journey, its flight would be neither accelerated nor maintained by the explosion of the charges it then might have left. To claim that it would be is to deny a fundamental law of dynamics, and only Dr. Einstein and his chosen dozen, so few and fit, are licensed to do that.

…..That Professor Goddard, with his "chair" in Clark College and the countenancing of the Smithsonian Institution, does not know the relation of action to reaction, and of the need to have something better than a vacuum against which to react--to say that would be absurd. Of course he only seems to lack the knowledge ladled out daily in high schools.

NYT, January 13, 1920

NYT, July, 1969


Chapter 9 20

Rockets

The rocket + exhaust gas has constant momentum in the absence of external forces (e.g. in outer space).

Constant Total Energy?

Constant Mechanical Energy?

Burning Fuel

Exhaust Gas

“The System”

Rocket


Chapter 9 21

Fundamentals of Physics

Chapter 9b Collisions1. Collisions2. Impulse & Linear Momentum3. Momentum & Kinetic Energy4. Inelastic Collisions in 1-Dimension5. Elastic Collisions in 1-Dimension6. Collisions in 2-Dimensions

Review & SummaryQuestionsExercises & Problems


Chapter 9 22

Collisions

A collision is an isolated event in which two or more bodies exert relatively strong forces on each other for a relatively short time.

A collision does not require physical contact.

Interacting

Moving freely

Moving freely

“Isolated” - no significant external forces during the collision.


Chapter 9 23

Impulse During a Collision

Newton’s 2nd Law:

During a collision, the left & right bodies briefly exert a force on each other:F(t) acts for time t

An Impulse

Newton’s 3rd Law

dt

PdFnet

(eq. 9-23)


Chapter 9 24

Impulse – Linear Momentum Theorem

(vectors!)

equal areas

Jppp if

dttFJ

tFJ avg


Chapter 9 25

A Series of Collisions

A stream of objects (e.g. bullets, a stream of water, …) hitting a target:

What is the force on the target?

v depends on what happens to the objects in the stream? Stop, bounce back,…

J

p

Fm

tvavg

vmvmnpnJ

tFJ avg


Chapter 9 26

Throw a ball against the wall J=? Favg = ?

a) Impulse on the ball?

b) Average force on the wall?

v v v v

v v v v vxf xi x

yf yi y

cos

sin sin

0

2

J p m v mv

J N s

2

18

sin

.

FJ

t

N s

sx Navg

18

0 0118 102.

..

030

01.010

0.6

3.0300

smst

v

kggm

sm


Chapter 9 27

Momentum & Kinetic Energy in Collisions

Collisions in a closed, isolated system:– “Closed” - no mass enters or leaves– “Isolated” - no external forces

– Elastic Collision• Total Kinetic Energy is conserved.• Total Linear momentum is conserved.

– Inelastic Collision• Total Linear momentum is conserved.

• Completely Inelastic Collision– The bodies stick together after they collide.

The linear momentum of each colliding body may change but the total momentum of the system cannot change, whether the collision is elastic or inelastic.


Chapter 9 28

Inelastic Collisions in One Dimension

afterbefore pp

ffii pppp 2121

ffii vmvmvmvm 22112211

momentum vectors:


Chapter 9 29

Completely Inelastic Collisions

stuck together

ivmm

mV 1

21

1

afterbefore pp

fii ppp 1221

Vmmmvm i

21211 0


Chapter 9 30

Velocity of the Center of mass

time

For an isolated system, the velocity of the center of mass cannot be changed by a collision:

Completely Inelastic Case:

vp p

m mcomi i

1 2

1 2

constant

ii ppP 21

comvMP

iicom vmvM

imM


Chapter 9 31

The Ballistic Pendulum

Velocity of a speeding bullet?

•Step 1: Bullet quickly stops in block

Completely inelastic collision.

Little time for block to move.

(Tension in cords approximately perpendicular to block’s direction of travel; no work done.)

vMm

mV

hgMmVMm 221

•Step 2: Bullet & block swing upward

Mechanical Energy Conserved.


Chapter 9 32

Colliding Blocks

Conservation of momentum: m v m v m v m vi i f f1 1 2 2 1 1 2 2

At maximum compression: v v vf f1 2

Conservation of total mechanical energy: K K Ui f spring

U k x 12

2

K m v m vi i i 12 1 1

2 12 2 2

2

K m m vf 12 1 2

2

algebra

xm m

k m mv vi i

1 2

1 21 2

m1 = 2.0 kg, 10 m/s

m2 = 5.0 kg, 3 m/s

k = 1120 N/m


Chapter 9 33

Elastic Collisions in One Dimension

In an elastic collision, the kinetic energy of each colliding body may change, but the total kinetic energy of the system does not change.

m v m v m v

m v m v m v

i f f

i f f

1 1 1 1 2 2

12 1 1

2 12 1 1

2 12 2 2

2

vm m

m mvf i1

1 2

1 21

vm

m mvf i2

1

1 21

2


Chapter 9 34

Elastic Collisions with a Moving Target

m v m v m v m vi i f f1 1 2 2 1 1 2 2

12 1 1

2 12 2 2

2 12 1 1

2 12 2 2

2m v m v m v m vi i f f

vm m

m mv

m

m mvf i i1

1 2

1 21

2

1 22

2

vm

m mv

m m

m mvf i i2

1

1 21

2 1

1 22

2


Chapter 9 35

Planetary Assist (“slingshot”)

vm m

m mv

m

m mvf i i1

1 2

1 21

2

1 22

2

m m

m m

m m

m

m m

2 1

1 2

1 2

2

1 2

1

22

v v v

v

v

f i i

f

fkm

s

1 1 2

1

1

2

12 2 13

38

v f1v = 12 km/s

vJ= 13 km/s


Chapter 9 36

Collisions in Two Dimensions

p p

p p p pbefore after

i i f f

1 2 1 2

For p m v m v m v

m v m vi f f

f f

21 1 1 1 1 1 2 2 2

1 1 1 2 2 2

0

0

: cos cos

sin sin


Chapter 9 37

Billiard Ball Collision

m v m v m v

v v v

i f f

i f f

1 1 2

1 1 2

12 1

2 12 1

2 12 2

2

12

12

22

m v m v m v

v v v

i f f

i f f

Pythagorean Theorem!

Equal mass elastic scattering.

e.g. billiards

090


Chapter 9 38

Equal Mass Elastic Collision

proton + proton proton + proton

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