chapter 7, 8 & 9 work and eergy - texas a&m...

Chapter 7, 8 & 9Chapter 7, 8 & 9Work and EergyWork and EergyProf. Rupak Mahapatra

Physics 218, Chapter 7 & 8 1

Checklist for Todayy•EOC Exercises from Chap 7 due on Monday

•Reading of Ch 8 due on Monday


Overview: Chapters 7, 8 & 9pCombine Chapter 7, 8 & 9 into six llectures

Today we’ll cover Work:y w W• The math I t iti d t di• Intuitive understanding

• Multiple ways to calculate workp yNext time:• How much energy does it take to • How much energy does it take to accomplish a task?


Why are we learning this stuff?y g

Thi i F nd m nt l t En in inThis is Fundamental to Engineering• How much work can a machine How much work can a machine do? (today)

• How much energy does it take to cc mplish t sk? (n xt tim )accomplish a task? (next time)

• Work and Energy relationshipWork and Energy relationship


The plan… p

Need to start with Need to start with some math some math…

S l d tScalar product


How do we Multiply Vectors?p y• First way: Scalar Product or Dot Product

– Why Scalar Product?•Because the result is a scalar •Because the result is a scalar (just a number)

– Why a Dot Product? •Because we use the notation A.BBecause we use the notation A B

A.B = |A||B|CosPhysics 218, Chapter 7 & 8 6

First Question:

B | ||B|C

Q

ˆˆA.B = |A||B|Cos?j i i Wh t

? i i is What

?j i is What


Examplep

ˆˆj A i A A YX

notation? Vector Unit using BA is Whatj B i B B YX

notation? Vector Unit using BA is What


Back to Work• The word “Work” means something gspecific in Physics (Kinda like Force)

• The amount of Work we do is the • The amount of Work we do is the amount of Forcing we do over some di tdistance

• Example: If we are accelerating a car p gfor 1 mile, then there is a force and a distance We do Workdistance We do Work


Calculating the workg• Work is done only if the force Wor s on on y f th forc (or some component of it) is in th m ( it )the same (or opposite)direction as the displacementdirection as the displacement

• Work is the force done P ll l t th di l tParallel to the displacement


Work for Constant ForcesThe Math: Work can be complicated.

h l Start with a simple caseDo it differently than the bookD ff yFor constant forces, the work is:

.(more on this later)


…(more on this later)

1 Dimension ExamplepYou pull a box with a constant force of p30N for 50m where the force and the displacement are in the same directiondisplacement are in the same direction

How much work is done on the box?W = F.d = 30N . 50m= 1500 N . M

= 1500 Joules 1500 Joules


What if the Force and the Displacement

h aren’t in the same di i ?direction?


2 Dim: Force Parallel to Displacement

W = F||d = F.d = Fdcos where is the ||angle between the net Force and the net displacement. You can think of this as the

Forceforce component in the direction of the displacement. ForceForce Rotate

DisplacementDisplacement F|| = Fcos


Work done and Work experiencedp•Something subtle: The amount of gwork YOU do on a body may not be same as the work done ON a bodysame as the work done ON a body

l h E f h •Only the NET force on the object is used in the total work calculation

Add ll th k d bj t •Add up all the work done on an object to find the total work done!


Examplesp• Holding a bag of groceries in placeplace– Is it heavy?–Will you get tired holding it?y g g–Are you doing “Work?”

• Moving a bag of groceries with g g gconstant speed across a room– Is it heavy?–Will you get tired doing it?–Are you doing “Work?”Lif i b f i • Lifting a bag of groceries a height h with constant speed

Work by you?Physics 218, Chapter 7 & 8 16

–Work by you?–Work on the bag?

Groceries: With the math• Holding a bag of groceries–W=F.d = Fdcos=(0)*(0)*cos = 0

• Moving a bag of groceries with constant speed across a room– Force exerted by you= mg, Net Force on bag = 0–Work on bag= F.d = Fdcos=0*dcos=0–Work exerted by you =Fdcos=mgd*cos(900)=0

• Lifting a bag of groceries a height h with constant speed–Work by you =Fdcos=(mg)hcos(00)=mgh–Work on bag = Fd*cos = (0)*h*(00) = 0


Work in Two DimensionsYou pull a crate of mass M a distance X along a h i t l fl ith t t f Y ll h horizontal floor with a constant force. Your pull has magnitude FP, and acts at an angle of . The floor is rough and has coefficient of friction is rough and has coefficient of friction .

Determine:• The work done by each force• The work done by each force• The net work on the crate

X


Checklist for Todayy•Finish Ch 7 this week

•Complete reading of Ch 8 by Monday class


What if the Force is What if the Force is changing direction?changing direction?

Wh t if th F i What if the Force is h i i d ?changing magnitude?


What if the force or direction isn’t constant?

I exert a force over a distance for awhile, then exert a different force over a different distance exert a different force over a different distance (or direction) for awhile. Do this a number of times. How much work did I do?

Need to add up all th littl the little pieces of pieces of

work!Physics 218, Chapter 7 & 8 21

Find the work: CalculusTo find the total work, we must sum up all the little pieces of work (i e F.d) If the force is continually pieces of work (i.e., F d). If the force is continually changing, then we have to take smaller and smaller lengths to add. In the limit, this sum becomes ang ,integral.

b

b

xdF

a

Total sum Int l

Physics 218, Chapter 7 & 8 22Integral

Use x-Integral for a Constant Force

Assume a constant Force, F, , ,doing work in the same direction, starting at x=0 and continuing for starting at x=0 and continuing for a distance d. What is the work?

Fd0FFd| FxFdx x dFW dxdd

Fd0FFd| FxFdx x dFW 0x00

Region of integrationRegion of integrationW=Fd


Non-Constant Force: Springsp g•Springs are a good

l f th t f example of the types of problems we come back pto over and over again!

•Hooke’s LawHooke s LawxkF

Some constantDisplacement

•Force is NOT Displacement

Physics 218, Chapter 7 & 8 24CONSTANT over a distance

Work done to stretch a Springp gHow much work do you do to stretch a spring stretch a spring (spring constant k) at constant k), at constant velocity (pulled D

slowly), from x=0 to x=D? t D?


Quiz: HikerQA hiker (mass M) carries a

backpack of mass m with constant speed up a hill of angle and height h.

D t iDetermine:

• The work done by theThe work done by the hiker

• The work done by gravity

• The work on the backpackPhysics 218, Chapter 7 & 8 26

• The work on the backpack

Kinetic Energy and Work-Energygy gy• Energy is another big concept in physics• If I do work, I’ve expended energy

– It takes energy to do work (I get t ta n rgy t w r ( g t tired)

• If net work is done on a stationary box it If net work is done on a stationary box it speeds up. It now has energy

• We say this box has “kinetic” energy! Think We say this box has kinetic energy! Think of it as Mechanical Energy or the Energy of Motion M

Kinetic Energy = ½mV2


Work Energy Relationshipgy p• If net positive work is done on a stationary p y

box it speeds up. It now has energy

•Work Equation naturally Work Equation naturally leads to derivation of kinetic energy

Kinetic Energy = ½mV2Physics 218, Chapter 7 & 8 28

Work-Energy Relationshipgy p•If net work has been done on an object, then it has a change in its kinetic energy (usually this means kinetic energy (usually this means that the speed changes)E i l t t t m t: If th i •Equivalent statement: If there is a change in kinetic energy then there has been net work on an objectCan use the change in energy to Can use the change in energy to

calculate the workPhysics 218, Chapter 7 & 8 29

Summary of equationsy q

Kinetic Energy = ½mV2Kinetic Energy = ½mV2

W= KEW= KECan use change in speed Can use change in speed to calculate the work, or ,the work to calculate the

speedPhysics 218, Chapter 7 & 8 30speed

Recipe to find work donep

Multiple ways to Multiple ways to calculate the work calculate the work

doneMultiple ways to Mu t p ways to

calculate the velocityy


Multiple ways to calculate workp y1. If the force and direction is constant

– F.d2 If the force isn’t constant or the 2. If the force isn t constant, or the

angles changeI– Integrate

3. If we don’t know much about the forces–Use the change in kinetic energy–Use the change in kinetic energy


Multiple ways to calculate velocityp y yIf we know the forces:•If the force is constant

F m V V t V2 V 2 2 d–F=ma →V=V0+at, or V2-V02 = 2ad

•If the force isn’t constant f f–Integrate the work, and look at th h i ki ti the change in kinetic energyW= KE = KEf-KEi W= KE = KEf KEi

= ½mVf2 -½mVi

2


Problem Solvingg

How do you solve Work How do you solve Work and Energy problems?and Energy problems?BEFORE and AFTERBEFORE and AFTER

Di mDiagrams


Problem SolvinggBefore and After diagramsfor an ft r agrams1.What’s going on beforeg g

work is done 2 Wh t’ i ft2.What’s going on after

work is donework is doneLook at the energy before and gythe energy after


Before…


After…


Conservation of Energy/ForcegyConservative and Non-conservative Forces• All forces can be divided in to 2 categories: whether work done depends categories whether work done depends on the path taken from point 1 to 2

• Conservative Force• Conservative Force– Work done is independent of the path– Such as Gravitational Force

• Non-conservative ForceNon conservative Force– Work done is dependent on the pathSu h s f i ti n l f

Physics 218, Chapter 7 & 8 38– Such as frictional force

Conservative Forces• Physics has the same meaning. Except

t ENFORCES th ti It’ t nature ENFORCES the conservation. It’s not optional, or to be fought for.

“A f i ti if th k d b “A force is conservative if the work done by a force on an object moving from one point to another point depends only on the initial and another point depends only on the initial and final positions and is independent of the particular path taken”particular path taken

• (We’ll see why we use this definition later)• (We ll see why we use this definition later)


Closed LoopspAnother definition:A force is conservative if the net work done by the force on an object moving around j gany closed path is zero

This definition and the previous one give the previous one give the same answer. Why?


Is Friction a Conservative Force?Force?


Checklist for Todayy•Quiz today


Quiz: Compressing a SpringQ p g p gA horizontal spring has spring

c nst nt kconstant k1.How much work must you do

to compress it from its to compress it from its uncompressed length (x=0) to a distance x=-D with no

l i ?acceleration?2.You then place a block of

mass m against the mass m against the compressed spring. Then you let go. Assuming no y g gfriction, what will be the speed of the block when it separates at x=0?


separates at x=0?

Potential Energygy• Things with potential: COULD do g pworkE G it ti t ti l • E.g. Gravitation potential energy:–If you lift up a brick it has the If you lift up a brick it has the potential to do damage


Example: Gravity & Potential Energy

You lift up a brick (at rest) from the d d h h ld i h i h h ground and then hold it at a height h

• How much work has been done on the brick?

• How much work did you do?y• If you let it go, how much work will be done by gravity by the time it hits the done by gravity by the time it hits the ground?

We say it has potential energy: We say it has potential energy: U=mgh

Physics 218, Chapter 7 & 8 45–Gravitational potential energy

Mechanical Energygy

•We define the total We define the total mechanical energy in a gysystem to be the kinetic n r plus th p t nti l energy plus the potential

energyenergy•Define E≡K+UDefine E K U


Conservation of Mechanical Energygy• For some types of problems, Mechanical

E i d ( thi t Energy is conserved (more on this next week)

• E.g. Mechanical energy before you drop a brick is equal to the mechanical energy q gyafter you drop the brick

K2+U2 = K1+U1Conservation of Mechanical EnergyConservation of Mechanical Energy

E2=E1


Problem Solvingg• What are the types of examples we’ll encounter?– Gravityy– Things fallingSprings– Springs

• Converting their potential energy into k d b k kinetic energy and back again

E = K + U = ½mv2 + mgyE = K + U = ½mv + mgy


Problem SolvinggFor Conservation of Energy problems: f gy p

BEFORE and AFTER diagramsg


QuizQ

We drop a ball from a We drop a ball from a height D above the groundg g

Using Conservation of E h i h d Energy, what is the speed just before it hits the just before it hits the ground?ground?


Potential EnergygyA brick held 6 feet in the air has potential energy

•Subtlety: Gravitational potential energy is l ti t h !relative to somewhere!

Example: What is the potential energy of a book 6 feet above a 4 foot high table? 10 feet above the feet above a 4 foot high table? 10 feet above the floor?

• U = U2-U1 = W = mg (h2-h1)U = U2-U1 = Wext = mg (h2-h1)•Write U = mghU=mgh + Const•U=mgh + Const

Only change in potential energy is really i f l

Physics 218, Chapter 7 & 8 51meaningful

Other Potential Energies: Springsg p g

Last week we calculated that it took ½kx2 of work to ½kx of work to compress a spring by a distance xdistance xHow much potential

d it h energy does it now how have?

U(x) = ½kx2


Problem SolvinggFor Conservation of Energy For Conservation of Energy problems:

BEFORE and AFTER diagramsg


QUIZ: Falling onto a SpringBefore After

Q g p gWe want to measure the spring constant of a certain spring. We drop a ball of k f k Z Zknown mass m from a known height Z above the uncompressed spring

C

uncompressed spring. Observe it compresses a distance C.What is the spring constant?


QUIZ: Bungee JumpQ g pA jumper of mass m

sits on a platform attached to a bungee lcord with spring constant k. The cord

l

constant k. The cord has length l (it doesn’t stretch untildoesn t stretch until it has reached this l th)length). How far does the

Physics 218, Lecture XII 55cord stretch y?

QUIZ: Friction and SpringsQ p gA block of mass mi liis traveling on a rough surface. It reaches a spring (spring constant k) ( p g )with speed vo and compresses it by ancompresses it by an amount D. D t iDetermine

Physics 218, Lecture XII 56

Roller Coaster You are in a roller coaster car of mass M

th t t t t th t h i ht Z ith that starts at the top, height Z, with an initial speed V0=0. Assume no friction.

) Wh t i th d t th b tt ?a) What is the speed at the bottom?b)How high will it go again?c) Would it go as high if there were friction?

ZZ


Non-Conservative Forces• In this problem there are three different types of forces actin :types of forces acting:1.Gravity: Conserves mechanical

energy2.Normal Force: Conserves

mechanical energy3 Friction: Doesn’t conserve 3.Friction: Doesn t conserve

mechanical energy•Since Friction causes us to lose mechanical •Since Friction causes us to lose mechanical energy (doesn’t conserve mechanical energy) it is a Non-Conservative force!


energy) it is a Non Conservative force!

Law of Conservation of Energygy• Mechanical Energy NOT always

dconserved• If you’ve ever watched a roller f y u wcoaster, you see that the friction turns the energy into heating the rails, the energy into heating the rails, sparks, noise, wind etc.

• Energy = Kinetic Energy + Potential • Energy = Kinetic Energy + Potential Energy + Heat + Others…T t l E i h t i –Total Energy is what is conserved!


Conservative ForcesIf there are only conservative forces in the problem,

h h i i f h i l then there is conservation of mechanical energy• Conservative: Can go back and forth along any path

d th t ti l d ki ti k and the potential energy and kinetic energy keep turning into one another

G d x mpl : G it nd Sp in– Good examples: Gravity and Springs• Non-Conservative: As you move along a path, the

potential energy or kinetic energy is turned into potential energy or kinetic energy is turned into heat, light, sound etc… Mechanical energy is lost.

Good example: Friction (like on Roller – Good example: Friction (like on Roller Coasters)


Law of Conservation of Energygy• Even if there is friction, Energy is conserved• Friction does work

– Can turn the energy into heatCan turn the energy into heat– Changes the kinetic energy

T l E K E P l •Total Energy = Kinetic Energy + Potential Energy + Heat + Others…– This is what is conserved

• Can use “lost” mechanical energy to estimate Can use lost mechanical energy to estimate things about friction


Roller Coaster with FrictionA roller coaster of mass m starts at rest

h h d f ll d h h h at height y1 and falls down the path with friction, then back up until it hits height py2 (y1 > y2). Assuming we don’t know anything about the Assuming we don t know anything about the friction or the path, how much work is done by friction on this path?done by friction on this path?


Checklist for Todayy•Ch 8 and 9 exercises due before Monday

•Read ch10 before Monday’s class


Energy Summarygy yIf there is net work on an object, it changes

the kinetic energy of the object (Gravity the kinetic energy of the object (Gravity forces a ball falling from height h to speed up Work done )up Work done.)

Wnet = KIf there is a change in the potential energy,

some one had to do some work: (Ball falling f h i ht h d k d l from height h speeds up→ work done → loss of potential energy. I raise a ball up, I do work which turns into potential energy for work which turns into potential energy for the ball)

U W WPhysics 218, Lecture XII 64

UTotal = WPerson =-WGravity

Energy Summarygy yIf work is done by a non-conservative force

it does negative work (slows something down), and we get heat, light, sound etc.

EHeat+Light+Sound.. = -WNCIf k i d n b n n n ti f If work is done by a non-conservative force,

take this into account in the total energy. (F i ti m h i l t b (Friction causes mechanical energy to be lost)

K1+U1 = K2+U2+EHeat… K +U = K +U W

Physics 218, Lecture XII 65K1+U1 = K2+U2-WNC

Force and Potential EnergygyIf we know the potential energy, U, we

f d h fcan find the force

dU dxx This makes sense… For example, the p ,force of gravity points down, but the potential increases as you go up

Physics 218, Lecture XIII 66

p y g p

Force and Potential Energygy

Draw some examplesDraw some examples…Gravity–Gravity–Spring


Mechanical Energygy

•We define the total We define the total mechanical energy in a gysystem to be the kinetic n r plus th p t nti l energy plus the potential

energyenergy•Define E≡K+UDefine E K U


Conservation of Mechanical Energygy• For some types of problems, Mechanical

E i d ( thi t Energy is conserved (more on this next week)

• E.g. Mechanical energy before you drop a brick is equal to the mechanical energy q gyafter you drop the brick

K2+U2 = K1+U1Conservation of Mechanical EnergyConservation of Mechanical Energy

E2=E1


Friction and Springsp gA block of mass mi t li is traveling on a rough surface. It reaches a spring (spring constant k) ( p g )with speed Vo and compresses it a compresses it a total distance D. D t min Determine

Physics 218, Lecture XV 70

Robot ArmA robot arm has a funny Force equation in 1Force equation in 1-dimension

23x

2

00X x

3x1F F

where F0 and X0 are

0x

0 0constants. The robot picks up a block at X=0 (at rest) and throws it (at rest) and throws it, releasing it at X=X0.What is the speed of


What is the speed of the block?

Potential Energy Diagramsgy g• For Conservative forces

d can draw energy diagrams

• Equilibrium points• Equilibrium points– Motion will move “around” the around the equilibriumIf placed there with – If placed there with no energy, will just stay (no force) 0F dUstay (no force) 0F dx

dUx


A football is thrownA football of mass m starts at rest and is

th ith d f 0 thrown with a speed of v0.

1. What is the final kinetic energy?2. How much work was done to reach this

velocity?

We don’t know the forces exerted by the arm as a function of time, but this allows us to sum them all up to calculate the work


Problem 1


Problem 1, continued


Problem 3


Problem 3, continued


Unknown Force


chapter 7, 8 & 9 work and eergy - texas a&m...

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