04/19/23 PHY 113 -- Lecture 9 1

Announcements

1. Exam 1 will be returned at the end of class. Please rework the exam, to help “solidify” your knowledge of this material. (Up to 10 extra credit points granted for reworked exam – turn in old exam, corrections on a separate paper – due Tuesday, Oct. 7.)

2. Physics colloquium today – 4 PM Olin 101

SPS meeting at 11:30 AM (free pizza for physics majors and potential physics majors)

3. Today’s topic – Chapter 9 HRW

Center of mass

Definition of momentum

Conservation of momentum

04/19/23 PHY 113 -- Lecture 9 2

Exam 1 -- PHY 113 D

0

1

2

3

4

5

50 55 60 65 70 75 80 85 90 95 100

Grade

04/19/23 PHY 113 -- Lecture 9 3

of mass

rirj

x

y

ii

iii

m

m rrCOM

04/19/23 PHY 113 -- Lecture 9 4

m1=1.2kg

m2=2.5kg

m3=3.4kg

321

332211

321

332211

mmmymymym

y

mmmxmxmxm

x

com

com

Example:

04/19/23 PHY 113 -- Lecture 9 5

Another example of center of mass:

x

y

(1m,2m)

(2m,1m)M0

4M0

m

mMMM

MMM

com

ji

jir

2.18.1

5142

5241

0

00

0

00

04/19/23 PHY 113 -- Lecture 9 6

ii

iii

com m

m aa

Position of the center of mass:

Velocity of the center of mass:

Acceleration of the center of mass:

ii

iii

com m

m rr

ii

iii

com m

m vv

04/19/23 PHY 113 -- Lecture 9 7

i

i

i

ii

iii

ii dt

ddtdm

mpv

aF

Physics of composite systems:

Center-of-mass velocity:

Note that:

M

m

m

mi

ii

ii

iii

com

vv

v

dtd

Mtotali

icomv

FF

04/19/23 PHY 113 -- Lecture 9 8

04/19/23 PHY 113 -- Lecture 9 9

A new way to look at Newton’s second law:

dtd

dtmd

dtd

mmpvv

aF )(

Define linear momentum p = mv

Consequences:

1. If F = 0 p = constant

2. For system of particles:

0dtdp

i

i

ii dt

dpF

i

ii

i

ii dt

dconstant 0 0 If p

pF

04/19/23 PHY 113 -- Lecture 9 10

Conservation of (linear) momentum:

If , p = constant 0Fpdtd

Example:

Suppose a molecule of CO is initially at rest, and it suddenly decomposes into separate C and O atoms. In this process the chemical binding energy E0 is transformed into mechanical energy of the C and O atoms. What can you say about the motion of these atoms after the decomposition?

Before After

vC

vO

mCvC-mOvO=0

vC=vO mO/mC

04/19/23 PHY 113 -- Lecture 9 11

Further analysis:

E0 = ½ mCvC2 + ½ mOvO

2 = ½ mCvC2 (1 + mC/mO)

½ mCvC2 = E0/(1 + mC/mO) ~ E0/(1 + 12/16) = 4/7 E0

½ mOvO2 = 3/7 E0

Extra credit opportunity:

Work through the details of the above analysis and verify the results for yourself ( perhaps with a different diatomic molecule).

04/19/23 PHY 113 -- Lecture 9 12

Peer instruction question:

Suppose a nucleus which has an initial mass of Mi= 238 m0 (where m0 denotes a standard mass unit),suddenly decomposes into two smaller nuclei with M1 = 234 m0 and M2 = 4 m0.

If the velocity of nucleus #1 is V1 what is the velocity of nucleus #2?

(a) -0.017V1 (b) -V1 (c) -59.5V1

After the decomposition which nucleus has more energy?

(a) M1 (b) M2 (c) The nuclei have the same energy.

(d) Not enough information is given.

04/19/23 PHY 113 -- Lecture 9 13

Peer instruction question:

Romeo (60 kg) entertains Juliet (40 kg) by playing his guitar from the rear of their boat (100 kg) which is at rest in still water. Romeo is 2m away from Juliet who is in the front of the boat. After the serenade, Juliet carefully moves to the rear of the boat (away from shore) to plant a kiss on Romeo’s cheek. How does the boat move relative to the shore in this process? (Initially Juliet is closest to the shore.)

(a) 0.2 m away from shore (b) 0.4 m away from shore

(c) 0.2 m toward shore (d) 0.4 m toward shore

x

CM

x’ d

CM

x-x’= d mJ/Mtotal = 0.4 m

04/19/23 PHY 113 -- Lecture 9 14

Another example:

In this case, mechanical energy is not conserved. (Where does it go?) However, to our level of approximation, we will assume that momentum is conserved.

Before: After:

mcv0c + mTv0T = mcvc + mT v

04/19/23 PHY 113 -- Lecture 9 15

Note: In general, momentum is a vector quantity.

mcv0c + mTv0T = (mc+ mT) vf

jiv TTc

Tc

Tc

cf v

mmm

vmm

m00

04/19/23 PHY 113 -- Lecture 9 16

04/19/23 PHY 113 -- Lecture 9 17

Statement of conservation of momentum:

inφinθ0

φcosθcos

2211

221111

svmsvm

vmvmvm

ff

ffi

If mechanical (kinetic) energy is conserved, then:2222

12112

12112

1ffi vmvmvm

04/19/23 PHY 113 -- Lecture 9 18

Peer instruction question:

Given the previous example, summarized with these equations:

which of the following statements are true?

(a) It is in principle possible to solve the above equations uniquely.

(b) It is not possible to solve the above equations uniquely because the mathematics is too difficult.

(c) It is not possible to solve the above equations uniquely because there is missing physical information.

inφinθ0

φcosθcos

2211

221111

svmsvm

vmvmvm

ff

ffi

2222

12112

12112

1ffi vmvmvm

04/19/23 PHY 113 -- Lecture 9 19

One dimensional case:

Conservation of momentum: m1v1i + m2v2i = m1v1f+m2v2f

Conservation of energy:

½ m1v1i2 + ½ m2v2i

2 = ½ m1v1f2+ ½ m2v2f

2

Extra credit: Show that

iif

iif

vmmmm

vmmm

v

vmmm

vmmmm

v

221

121

21

12

221

21

21

211

2

2

04/19/23 PHY 113 -- Lecture 9 20

dtdp

F

Linear momentum: p = mv

Generalization for a composite system:

i

i

ii dt

dpF

)constant(;0;0If i

ii

i

ii dt

dp

pF

Conservation of momentum:

Energy may also be conserved ( for example, in an “elastic” collision)

Summary

04/19/23 PHY 113 -- Lecture 9 21

Another example

h1

1. m1 falls a distance h1 – energy conserved

2. m1 collides with m2 – momentum and energy conserved

3. m1 moves back up the incline to a height h1’

h1’

Extra credit:

Show that 1

2

21

211 h

mmmm

h

04/19/23 PHY 113 -- Lecture 9 22

Notion of impulse:

Ft = pdtdp

F

Example:

vi

vf

mi

f

p = mvf - mvi

p=(mvf sin f + mvf sin f) i

+(-mvf cos f + mvf cos f) j

pfpi

04/19/23 PHY 113 -- Lecture 9 23

Notion of impulse:

dtdp

F

Example:

vi

vf

mi

f

p = mvf - mvi

p=(mvf sin f + mvf sin f) i

+(-mvf cos f + mvf cos f) j

pfpi

p = pf - pi = F t

cause effect