Physics 121. Tuesday, February 26,...

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Physics 121.Tuesday, February 26, 2008.


Physics 121.Tuesday, February 26, 2008.

• Course Information

• Quiz

• Topics to be discussed today:

• Review of Conservation Laws (kinetic energy, potential energy,conservative and con-conservative forces).

• Dissipative forces.

• Gravitational potential energy.


Course information.Exam # 1.

• On Thursday February 28 between 8 am and 9.30 am thefirst midterm exam of Physics 121 will be held:• Material covered: Chapters 1 - 6 of our text book.• Location: Hubbell.

• There will be a normal lecture after the exam (at 9.40 am inHoyt).

• A Q&A session on the material covered on exam # 1 willtake place on Tuesday evening 2/26 between 9 pm and 11pm in Hoyt (location needs to be confirmed).

• There will be extra office hours on Tuesday 2/26 andWednesday 2/28:• Tuesday: 1 - 4 pm (B&L 304) + 5 - 9 pm (POA library, 2 TAs)• Wednesday: 1 - 3 pm (B&L 304) + 2.45 - 4.45 pm (POA library, 1

TA) + 7 - 10 pm (2 - 3 TAs)


Course information.Exam # 1.

• During workshops on Tuesday 2/26 and Wednesday 2/27,the focus will be exam # 1. You can attend any (or all)workshops on these days. Bring your questions!

• There will be no workshops and office hours on Thursday2/28 and Friday 2/29.

• You will receive the exam back during workshop during theweek of March 3.

• Any corrections to the grades of your grade can only bemade by me, not by your TAs.

• The TAs will not see the exam until you see it.


Physics 121.Quiz lecture 11.

• The quiz today will have 3 questions.


Conservation of energy.A review.

• The mechanical energy of a system is defined as the sum ofthe kinetic energy K and the potential energy U:

E = K + U

• If the total mechanical energy is constant, we must requirethat ΔE = 0, or

ΔK + ΔU = 0

• We conclude that any change in the kinetic energy ΔK mustbe accompanied by an equal but opposite change in thepotential energy ΔU.



• Per definition, the change inpotential energy is related to thework done by the force.

• The difference between thepotential energy at (2) and at (1)depends on the work done by theforce F along the path between(1) and (2).

• The potential at (2) is onlyuniquely defined if the work doneby the force is independent of thepath.



• If the work is independent of thepath, the work around a closedpath will be equal to 0 J.

• A force for which the work isindependent of the path is calleda conservative force.

• A force for which the workdepends on the path is called anon-conservative force.



• Examples of conservative forces:• The spring force• The gravitational force

• Note: the conservative force issometimes directed in thedirection of motion, sometimes inthe opposite direction.

• Examples of non-conservativeforces:

• The kinetic friction force• The drag force


Conservative forces.Path independence.

• The work done by anyconservative force depends onlyon the start end end points, and isindependent of the path followed.

• Let’s proof this for thegravitational force when wemove from A to B.

• We will consider two paths:

• Directly from A to B (along theline connecting A and B.

• Horizontal motion followed byvertical motion.

1

θ

mgmg

mg

A

B

2

d

h


Conservative forces.Path independence.

• Path 1 (direct route):

• Path 2 (D tour):The work done by thegravitational force when youmove from A in the horizontaldirection is zero (path and forceare perpendicular). The workdone by the gravitational forceduring the vertical segment isequal to

1

θ

mgmg

mg

A

B

2

d

h

W1=!Fi

!d = mgd cos

1

2! +"#

$%&'(= )mgd sin"

W1=!Fi

!d = mghcos 180°( ) = !mgd sin"



• Applying conservation ofmechanical energy usually simplifiesour calculations.

• However, it can only be used if wecare only about the relation betweenthe initial state of a system and thefinal state.

• For example, conservation ofmechanical energy will tell usimmediately that the two kids on thetwo slides will have the samevelocity at the bottom of the slide.But we can not say anything abouttheir relative time of arrival.



• Let’s test our understanding of the concepts of mechanicalenergy and work by working on the following conceptproblems:

• Q11.1

• Q11.2

• Q11.3

• Q11.4

• Note: for each of these 4 problems I like to get just youropinion, not a group opinion.


Conservation of energy.Dissipative forces.

• When dissipative forces, such as friction forces, are present,mechanical energy is no longer conserved.

• For example, a friction force will reduce the speed of amoving object, thereby dissipating its kinetic energy.

• The amount of energy dissipated by these non-conservativeforces can be calculated if we know the magnitude anddirection of these forces along the path followed by theobject we are studying:

ΔK + ΔU = WNC

where WNC is the work done by the non-conservative forces.


Conservation of energy.Dissipative forces.

• When dissipative forces arepresent, some or all of themechanical energy is convertedinto “internal” energy.

• The internal energy is usually inthe form of heat.

Bouncing a ball (credit: NASA)Sharpening a pencil (credit: NASA)


Problems with dissipative forces.An example.

• A ball bearing whose mass is mis fired vertically downward froma height h with an initial velocityv0. It buries itself in the sand at adepth d. What average upwardresistive force f does the sandexert on the ball as it comes torest?

• The initial total mechanicalenergy of the system is equal to

m

vo

h

df

y = 0U = 0

Ei =Ui + Ki = mgh +1

2mv

0

2



• The final total mechanical energyof the system is equal to just thepotential energy of the bearingwhich is equal to Ef = -mgd.

• Mechanical energy is notconserved since the friction forcedissipates some of the mechanicalenergy. The work done by thefriction force is equal to Wf = -fd.

• The change in the mechanicalenergy of the system is

m

vo

h

df

y = 0U = 0

!E = Ef " Ei = "mgd " mgh "1

2mv

0

2= "mg h + d( ) "

1

2mv

0

2



• The change in the mechanicalenergy of the system is must beequal to the work done by thefriction force (assuming this isthe only dissipative force actingon the system):

• This relation can now be used tocalculate the magnitude of thefriction force f:

m

vo

h

df

y = 0U = 0

! fd = !mg h + d( ) !1

2mv

0

2

f = mg + mgh

d+1

2

mv0

2

d


One final application.The gravitational potential energy.

• In our discussion so far we havetaken the gravitational potentialenergy to be equal to mgh.

• This is correct when we are veryclose to the surface of the earth,but not correct when we are a fewmiles from the surface of theearth.

• The work done by thegravitational force when wemove an object from (1) to (2) isequal to

W = GmME(1/r2 - 1/r1)



• The potential energy differencebetween position (1) and (2) is thusequal to

U2 - U1 = -W = GmME(-1/r2+1/r1)

• It is common to choose thegravitational potential energy to beequal to 0 J at infinity. With thechoice, the potential energy at (1)will be equal to

U1 = -GmME/r1



• Consider an object on the surface ofthe earth with a kinetic energy K(rE).The total mechanical energy of theobject is U(rE) + K(rE).

• If U(rE) + K(rE) < 0 then there will bea distance r where U(r) = U(rE) +K(rE). At that distance K(r) = 0 J.The object can not escape!

• If U(rE) + K(rE) > 0 then at everydistance r, K(r) > 0 J. The objectcan escape!

• The limiting case occurs when U(rE)+ K(rE) = 0. The velocity for whichthis is case, it called the escapevelocity. For the earth, this velocityis 11,200 m/s.

“Bound” to the earth.

Not “bound” to the earth.


Done for today!Thursday: conservation of linear momentum.

Physics 121. Tuesday, February 26,...

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