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12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 1

PHY 113 C General Physics I11 AM – 12:15 PM MWF Olin 101

Plan for Lecture 26:

1. Comments on preparing for Final Exam2. Comprehensive review – Part II3. Course assessment

12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 2

12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 3

Final exam schedule for PHY 113 C

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12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 4

Comments on Final Exam It will be comprehensive (covering material

from Chapters 1-22) It is scheduled for 9 AM Dec. 12th in Olin 101 In class format only; no time pressure May bring 4 equation sheets Format will be similar to previous exams; may

see problems similar to those on previous exams

12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 5

General advice on how to prepare for Final Exam

Review fundamental concepts and their corresponding equations

Develop equation sheets that help you solve example problems on all of the material. (You can assume that empirical constants and parameters will be given to you; they need not take up space on your equation sheet.)

Practice problem solving techniques. If you find mysteries, unanswered questions, etc.,

please contact me.

12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 6

Problem solving steps

1. Visualize problem – labeling variables2. Determine which basic physical principle(s) apply3. Write down the appropriate equations using the

variables defined in step 1.4. Check whether you have the correct amount of

information to solve the problem (same number of knowns and unknowns).

5. Solve the equations.6. Check whether your answer makes sense (units,

order of magnitude, etc.).

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12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 7

Review of some basic concepts

Vectors Keep track of 2 or

more components (or magnitude and direction)

Examples Position vector Velocity Acceleration Force Momentum

Scalars Single (signed)

quantity Examples Time Energy Kinetic energy Work Potential energy Pressure Temperature Mass Density Volume

12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 8

Review of some basic concepts

Newton’s second law

Fpv

FrvFa

dtd

dtmd

dtdm

dtdm

m

2

2

system)extendedofmassofcenter (or particlepoint singleFor

ii

i

i

ii

iii

dtd

m

Fp

FaparticlesofsystemFor

12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 9

Review of some basic concepts

Newton’s second law for angular motion

τFrLprpr

Fp

dtd

dtd

dtd

dtd

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12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 10

rFr

r

dWf

i

fi : workofDefinition

Review of energy concepts:

2

21:energykineticofDefinition mvK

22

21

21

:oremenergy thekinetic-Work

if

f

itotal

totalfi mvmvdW rF

12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 11

22

21

21

:oremenergy thekinetic-Work

if

f

itotal

totalfi mvmvdW rF

Summary of work, potential energy, kinetic energy relationships

edissipativ

fiiiff

ifedissipativ

fiiftotal

fi

WUKUK

KKWUUW

:gRearrangin

rr

edissipativfiif

edissipativfi

veconservatifi

totalfi

WUU

WWW

12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 12

Extension of concepts of energy conservation to extended objects

rotationtotal KKK massofcenter

energyKinetic

edissipativfiiiff WUKUK

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12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 13

22

21

21

:objectrollingofenergy kineticTotal

CM

CMrollingtotal

MvI

KKK

CMvRdtdR

dtds

dtd

: thatNote

22

222

21

21

21

CM

CM

CMrollingtotal

vMRI

MvRRI

KKK

22

21

21

:objectrollingofenergy kineticTotal

CM

CMrollingtotal

MvI

KKK

CMvRdtdR

dtds

dtd

: thatNote

22

222

21

21

21

CM

CM

CMrollingtotal

vMRI

MvRRI

KKK

CMCM

12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 14

Three round balls, each having a mass M and radius R, start from rest at the top of the incline. After they are released, they roll without slipping down the incline. Which ball will reach the bottom first?

AB C

2MRI A 22 5.0

21 MRMRIB

22 4.052 MRMRIC

2

22

/12

01210

MRIghv

vMR

IMMgh

UKUK

CM

CM

ffii

12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 15

iclicker exercise:In previous example which of the equations on your equation sheet would be most useful?

B&A C.

rollingFor ;21

21B.

A.

22 RvIMvK

UKUK

CMCMtotal

ffii

2/12D.

MRIghvCM

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12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 16

From your questions -- (question from Exam 2)

21

21

12221

2112 0ˆ

RRmGmU

RRmGm

τrF

12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 17

Comment on circular motion -- uniform circular motion

ra ˆ

:ison acceleratilcentripeta theanddirection radialin the

onaccelerati then the,If

2

rv

vvv

c

fi

12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 18

r

ra

ra

ra

ˆ2

ˆ2

ˆ

2

2

2

rf

rT

rv

c

c

c

Trv 2

In terms of time period T for one cycle:

In terms of the frequency f of complete cycles:πfrv

Tf 2;1

Comment on circular motion -- uniform circular motion

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12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 19

Comment on circular motion -- uniform circular motion –effects on gravitationally attractive bodies

221

21

1

21

1

1221

21111

12221

2112

ˆˆ

ˆ

RRmGm

Rvm

RRmGmam

RRmGm

RR

rF

221

21

2

1

2RR

GmRT

12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 20

Comment on circular motion -- non-uniform circular motion

r

ra

ra

ra

ˆ2

ˆ2

ˆ

2

2

2

rf

rT

rv

c

c

c

At each instant of time

Note that if speed v is not constant, then there will alsobe a tangential component of acceleration:

θa ˆdtdv

aca

12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 21

From your questions -- (question from Exam 1)

a. Neglecting any possible dissipative forces acting on this system, determine the magnitude of the velocity of the ball vf as it is caught by the person at the coordinates (xf,yf).

b. What is the angle f?c. Determine the net work of gravity on the

ball at it moves from the initial to final positions in its trajectory: .

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12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 22

From your questions -- (question from Exam 1) a. Neglecting any possible dissipative

forces acting on this system, determine the magnitude of the velocity of the ball vf as it is caught by the person at the coordinates (xf,yf).

b. What is the angle f?c. Determine the net work of gravity on the

ball at it moves from the initial to final positions in its trajectory: .

)(:gravityby Work (c)

(b)for for Solvecoscos:constantis velocity horizontal that Note

(a)for for Solve210

21

:energyofon conservatiusingSolution

22

if

fffii

fffi

ffii

yymgWvv

vmgymvmv

UKUK

12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 23

From your questions -- force diagrams

m

1 2

F1F2

mg

0sinsin0coscos

0:mequilibriuin systemFor

2211

2211

mgFFFF

ii

F

12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 24

mg(-j)

r

T F=ma

T- mg cos 0

mg sin ma

t=I a r mg sin = mr2 a mra

From your questions -- pendulum

rg

rg

dtd

dtdmr

dtmrdmgr

dtdLτ

sin:equationsPendulum

sin

:elyAlternativ

2

2

2

22

2

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12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 25

From your questions -- driven Harmonic oscillator

12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 26

From your questions -- driven Harmonic oscillator

t

mk

mFtmkAtx

tFkxdt

xdm

Fma total

sin/cos)(:solutionGeneral

sin

2

0

02

2

12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 27

Similar problem from webassign:

Damping is negligible for a 0.165-kg object hanging from a light, 6.30-N/m spring. A sinusoidal force with an amplitude of 1.70 N drives the system. At what frequency will the force make the object vibrate with an amplitude of 0.600m?

tmk

mFtmkAtx

sin/cos)(

2

0

(usually neglected)

mF

mk

mk

mF

6.0

6.0/:caseIn this

02

2

0

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12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 28

Examples of two-dimensional collision; balls moving on a frictionless surface

smsmsmv

smsmsm

vvvsmsm

vvvmvm

vmvmvmsmv

smvkgmm

oof

o

fif

o

ff

ff

ffi

f

i

/11.188.17cos/060.1

88.17sin/342.0

88.71060.1342.0tan

/060.120cos/1/2

coscos/342.020sin/1

sinsinsinsin0

coscos20,/1

,/2,06.0:Suppose

1

o

211

21

2211

221111

o2

121

12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 29

Examples of two-dimensional collision; balls moving on a frictionless surface – energy conservation?

Note: In these collision analyses, we are neglecting forces and potential energy

iclicker questionWhy?

A. We are cheating physicsB. We are applying the laws of

physics correctly

12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 30

Examples of two-dimensional collision; balls moving on a frictionless surface – energy conservation?

Assuming that we applying the laws of physics correctly – we can ask the question – Is (kinetic) energy conserved?

processin lost or addedEnergy IfconservedisEnergy If

21

21

021

222

211

211

fi

fi

fff

ii

KKKK

vmvmK

vmK

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12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 31

From your questions -- conservation of angular momentum

mm

d1 d1

mm

d2 d2

I1=2md12 I2=2md2

2

I11=I22 2=1 I1/I2

1 2

constantis then 0,If LττL

vrL

dtd

ILmi

iii

12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 32

Example form Webassign #11

X

t1

t3

t2

iclicker exerciseWhen the pivot point is O, which torque is zero?

A. t1?B. t2?C. t3?

12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 33

An example of the application of torque on a rigid object:

A horizontal 800 N merry-go-round is a solid disc of radius 1.50 m and is started from rest by a constant horizontal force of 50 N applied tangentially to the cylinder. Find the kinetic energy of solid cylinder after 3 s.

K = ½ I 2 t I a i at = atIn this case I = ½ m R2 and t = FR

R F

JsN

Nmg

tFgRItFt

IFRIIK

Rg

mgItI

FRtIFR

625.275)3(80050m/s8.9

/21

21

21

21

22

222

2

2222

2

aa

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12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 34

Webassign questions on fluids (Assignment #17)

A hypodermic syringe contains a medicine with the density of water (see figure below). The barrel of the syringe has a cross-sectional area A = 2.40 10-5 m2, and the needle has a cross-sectional area a = 1.00 10-8 m2. In the absence of a force on the plunger, the pressure everywhere is 1.00 atm. A force of magnitude 2.65 N acts on the plunger, making medicine squirt horizontally from the needle. Determine the speed of the medicine as it leaves the needle's tip.

122121

22222

111

212

1

;/;:caseIn this AvavAFPPyyPgyvPgyv

22

1

2

AaA

Fvv

12/05/2013 PHY 113 C Fall 2013 -- Lecture 26 35

Send email or come to see me if you have further questions.

THANKS!