Chapter 7, 8 & 9 Work and Eergy Prof. Rupak Mahapatra Physics 218, Chapter 7 & 8 1
Checklist for Today EOC Exercises from Chap 7 due on Monday Reading of Ch 8 due on Monday Physics 218, Chapter 7 & 8 2
Overview: Chapters 7, 8 & 9 Combine Chapter 7, 8 & 9 into six lectures Today ywe ll cover Work: The math Intuitive understanding di Multiple ways to calculate work Next time: How much energy does it take to accomplish a task? Physics 218, Chapter 7 & 8 3
Why are we learning this stuff? This is Fundamental Fndmntl to Enin Engineering How much work can a machine do? (today) How much energy does it take to accomplish a task? (next time) Work and Energy relationship Physics 218, Chapter 7 & 8 4
The plan Need to start with some math Scalar l product Physics 218, Chapter 7 & 8 5
How do we Multiply py Vectors? First way: Scalar Product or Dot Product Why Scalar Product? Because the result is a scalar (just a number) Why a Dot Product? Because we use the notation A. B A. B = A B Cos Physics 218, Chapter 7 & 8 6
First Question: A. B = A B Cos B C What is ˆî ˆî? What is îi ĵ j? Physics 218, Chapter 7 & 8 7
Example A AX î AY ĵ B B ˆ ˆ X i BY j What is A B using Unit Vector notation? Physics 218, Chapter 7 & 8 8
Back to Work The word Work means something specific in Physics (Kinda like Force) The amount of Work we do is the amount of Forcing we do over some distance Example: If we are accelerating a car for 1 mile, then there is a force and a distance We do Work Physics 218, Chapter 7 & 8 9
Calculating the work Work is done only if the force (or some component of it) is in the same (or opposite) direction as the displacement Work is the force done Parallel l to the displacement Physics 218, Chapter 7 & 8 10
Work for Constant Forces The Math: Work can be complicated. Start with a simple case Do it differently than the book For constant forces, the work is:. (more on this later) Physics 218, Chapter 7 & 8 11
1 Dimension Example You pull a box with a constant force of 30N for 50m where the force and the displacement are in the same direction How much work is done on the box? W = F. d = 30N. 50m= 1500 N. M = 1500 Joules Physics 218, Chapter 7 & 8 12
What if the Force and the Displacement aren t in the same direction? i Physics 218, Chapter 7 & 8 13
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 force component in the direction of the displacement. Force Rotate Force Displacement F = Fcos Displacement Physics 218, Chapter 7 & 8 14
Work done and Work experienced Something subtle: The amount of work YOU do on a body may not be same as the work done ON a body Only the NET force on the object is used in the total work calculation Add up all the work done on an object to find the total work done! Physics 218, Chapter 7 & 8 15
Examples Holding a bag of groceries in place Is it heavy? Will you get tired holding it? Are you doing Work? Moving a bag of groceries with constant speed across a room Is it heavy? Will you get tired doing it? Are you doing Work? Lifting i a bag of groceries a height h with constant speed Work by you? Work on the bag? Physics 218, Chapter 7 & 8 16
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(90 0 )=0 Lifting a bag of groceries a height h with constant speed Work by you =Fdcos =(mg)hcos(0 0 )=mgh Work on bag = Fd*cos = (0)*h*(0 0 ) = 0 Physics 218, Chapter 7 & 8 17
Work in Two Dimensions You pull a crate of mass M a distance X along a horizontal floor with a constant t force. Your pull has magnitude F P, and acts at an angle of. The floor is rough and has coefficient of friction. Determine: The work done by each force The net work on the crate X Physics 218, Chapter 7 & 8 18
Checklist for Today Finish Ch 7 this week Complete reading of Ch 8 by Monday class Physics 218, Chapter 7 & 8 19
What if the Force is changing direction? What if the Force is changing magnitude? Physics 218, Chapter 7 & 8 20
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 (or direction) for awhile. Do this a number of times. How much work did I do? Need to add up all the little pieces of work! Physics 218, Chapter 7 & 8 21
Find the work: Calculus To find the total work, we must sum up all the little 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 an integral. b a F dx Total sum Integral Physics 218, Chapter 7 & 8 22
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 a distance d. What is the work? d d x W F dx Fdx Fx x 0 0 0 d Fd F 0 Fd Region of integration W=Fd Physics 218, Chapter 7 & 8 23
Non-Constant Force: Springs Springs are a good example of the types of problems we come back to over and over again! Hooke s s Law F kx Some constant Displacement Force is NOT Physics 218, Chapter 7 & 8 24 CONSTANT over a distance
Work done to stretch a Spring How much work do you do to stretch a spring (spring constant k), at constant velocity (pulled slowly), from x=0 to x=d? D Physics 218, Chapter 7 & 8 25
Quiz: Hiker A hiker (mass M) carries a backpack of mass m with constant speed up a hill of angle and height h. Dt Determine: The work done by the hiker The work done by gravity The work on the backpack Physics 218, Chapter 7 & 8 26
Kinetic Energy and Work-Energy Energy is another big concept in physics If I do work, I ve expended energy It takes energy to do work (I get tired) If net work is done on a stationary box it speeds up. It now has energy We say this box has kinetic energy! Think of it as Mechanical Energy or the Energy of Motion Kinetic Energy = ½mV 2 Physics 218, Chapter 7 & 8 27
Work Energy Relationship If net positive work is done on a stationary box it speeds up. It now has energy Work Equation naturally leads to derivation of kinetic energy Kinetic Energy = ½mV 2 Physics 218, Chapter 7 & 8 28
Work-Energy Relationship If net work has been done on an object, then it has a change in its kinetic energy (usually this means that the speed changes) Equivalent E i l t statement: t t: If there is a change in kinetic energy then there has been net work on an object Can use the change in energy to calculate the work Physics 218, Chapter 7 & 8 29
Summary of equations Kinetic Energy = ½mV 2 W= KE Can use change in speed to calculate the work, or the work to calculate the speed speed Physics 218, Chapter 7 & 8 30
Recipe to find work done Multiple ways to calculate the work done Multiple ways to calculate the velocity Physics 218, Chapter 7 & 8 31
Multiple ways to calculate work 1. If the force and direction is constant F. d 2. If the force isn t constant, or the angles change Integrate 3. If we don t know much about the forces Use the change in kinetic energy Physics 218, Chapter 7 & 8 32
Multiple ways to calculate velocity If we know the forces: If the force is constant F=ma F m V=VV V 2 0 +at, or V 2 -V 02 = 2ad If the force isn t constant Integrate the work, and look at the change in kinetic energy W= KE = KE f -KE i = ½mV 2 - f ½mV 2 i Physics 218, Chapter 7 & 8 33
Problem Solving How do you solve Work and Energy problems? BEFORE and AFTER Diagrams Physics 218, Chapter 7 & 8 34
Problem Solving Before and After diagrams 1.What s going g on before work is done 2.What s going on after work is done Look at the energy before and the energy after Physics 218, Chapter 7 & 8 35
Before Physics 218, Chapter 7 & 8 36
After Physics 218, Chapter 7 & 8 37
Conservation of Energy/Force Conservative and Non-conservative Forces All forces can be divided in to 2 categories: whether work done depends on the path taken from point 1 to 2 Conservative Force Work done is independent of the path Such as Gravitational Force Non-conservative Force Work done is dependent on the path Such as frictional force Physics 218, Chapter 7 & 8 38
Conservative Forces Physics has the same meaning. Except nature ENFORCES the conservation. It s not optional, or to be fought for. 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 final positions and is independent of the particular path taken (We ll see why we use this definition later) Physics 218, Chapter 7 & 8 39
Closed Loops Another definition: A force is conservative if the net work done by the force on an object moving around any closed path is zero This definition and the previous one give the same answer. Why? Physics 218, Chapter 7 & 8 40
Is Friction a Conservative Force? Physics 218, Chapter 7 & 8 41
Quiz today Checklist for Today Physics 218, Chapter 7 & 8 42
Quiz: Compressing a Spring A horizontal spring has spring constant nt k 1.How much work must you do to compress it from its uncompressed length (x=0) to a distance x=-d with no acceleration? 2.You then place a block of mass m against the compressed spring. Then you let go. Assuming no friction, what will be the speed of the block when it separates at x=0? Physics 218, Chapter 7 & 8 43
Potential Energy Things with potential: COULD do work E.g. Gravitation ti potential ti energy: If you lift up a brick it has the potential to do damage Physics 218, Chapter 7 & 8 44
Example: Gravity & Potential Energy You lift up a brick (at rest) from the ground and then hold it at a height h h How much work has been done on the brick? How much work did you do? If you let it go, how much work will be done by gravity by the time it hits the ground? We say it has potential energy: U=mgh Gravitational potential energy Physics 218, Chapter 7 & 8 45
Mechanical Energy We define the total mechanical energy in a system to be the kinetic energy plus the potential energy Define E K+U Physics 218, Chapter 7 & 8 46
Conservation of Mechanical Energy For some types of problems, Mechanical Energy is conserved (more on this next week) E.g. Mechanical energy before you drop a brick is equal to the mechanical energy after you drop the brick K 2 +U 2 = K 1 +U 1 Conservation of Mechanical Energy E 2 =E 1 Physics 218, Chapter 7 & 8 47
Problem Solving What are the types of examples we ll encounter? Gravity Things falling Springs Converting their potential energy into kinetic energy and back again E = K + U = ½mv 2 + mgy Physics 218, Chapter 7 & 8 48
Problem Solving For Conservation of Energy problems: BEFORE and AFTER diagrams Physics 218, Chapter 7 & 8 49
Quiz We drop a ball from a height D above the ground Using Conservation of Energy, what is the speed just before it hits the ground? Physics 218, Chapter 7 & 8 50
Potential Energy A brick held 6 feet in the air has potential energy Subtlety: Gravitational potential energy is relative to somewhere! Example: What is the potential energy of a book 6 feet above a 4 foot high table? 10 feet above the floor? U = U 2 -U 1 = W ext = mg (h 2 -h 1 ) Write U = mgh U=mgh + Const Only change in potential energy is really meaningful Physics 218, Chapter 7 & 8 51
Other Potential Energies: Springs Last week we calculated that it took ½kx 2 of work to compress a spring by a distance x How much potential energy does it now how have? U(x) = ½kx 2 Physics 218, Chapter 7 & 8 52
Problem Solving For Conservation of Energy problems: BEFORE and AFTER diagrams Physics 218, Chapter 7 & 8 53
QUIZ: Falling onto a Spring We want to measure the spring constant of a certain spring. We drop a ball of known mass m from a known height Z above the uncompressed spring. Observe it compresses a distance C. What is the spring constant? Before Z After Z C Physics 218, Chapter 7 & 8 54
QUIZ: Bungee Jump A jumper of mass m sits on a platform attached to a bungee cord with spring constant k. The cord has length l (it doesn t stretch until it has reached this length). ll How far does the cord stretch y? Physics 218, Lecture XII 55
QUIZ: Friction and Springs A block of mass m is traveling on a rough surface. It reaches a spring (spring (p constant k) ) with speed v o and compresses it by an amount D. Determine t i Physics 218, Lecture XII 56
Roller Coaster You are in a roller coaster car of mass M that t starts t at the top, height ht Z, Z with an initial speed V 0 =0. Assume no friction. 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? Z Physics 218, Chapter 7 & 8 57
Non-Conservative Forces In this problem there are three different types of forces acting: 1.Gravity: Conserves mechanical energy 2.Normal Force: Conserves mechanical energy 3.Friction: Doesn t conserve mechanical energy Since Friction causes us to lose mechanical energy (doesn t conserve mechanical energy) it is a Non-Conservative force! Physics 218, Chapter 7 & 8 58
Law of Conservation of Energy Mechanical Energy NOT always conserved If fyu you ve ever watched a roller coaster, you see that the friction turns the energy into heating the rails, sparks, noise, wind etc. Energy = Kinetic Energy + Potential Energy + Heat + Others Total Energy is what ht is conserved! Physics 218, Chapter 7 & 8 59
Conservative Forces If there are only conservative forces in the problem, then there is conservation of mechanical energy Conservative: Can go back and forth along any path and the potential ti energy and kinetic energy keep turning into one another Good examples: Gravity and Springs Non-Conservative: As you move along a path, the potential energy or kinetic energy is turned into heat, light, sound etc Mechanical energy is lost. Good example: Friction (like on Roller Coasters) Physics 218, Lecture XII 60
Law of Conservation of Energy Even if there is friction, Energy is conserved Friction does work Can turn the energy into heat Changes the kinetic energy Total Energy = Kinetic Energy + Potential Energy + Heat + Others This is what is conserved Can use lost mechanical energy to estimate things about friction Physics 218, Lecture XII 61
Roller Coaster with Friction A roller coaster of mass m starts at rest at height h y 1 and falls down the path with friction, then back up until it hits height y 2 (y 1 > y 2 ). Assuming we don t know anything about the friction or the path, how much work is done by friction on this path? Physics 218, Lecture XII 62
Checklist for Today Ch 8 and 9 exercises due before Monday Read ch10 before Monday s class Physics 218, Chapter 7 & 8 63
Energy Summary If there is net work on an object, it changes the kinetic energy of the object (Gravity forces a ball falling from height h to speed up Work done.) W net = K If there is a change in the potential energy, some one had to do some work: (Ball falling from height ht h speeds up work done loss of potential energy. I raise a ball up, I do work which turns into potential energy for the ball) U Total = W Physics Person =-W 218, Lecture XII Gravity 64
Energy Summary If work is done by a non-conservative force it does negative work (slows something down), and we get heat, light, sound etc. E Heat+Light+Sound.. = -W NC If work is done by a non-conservative n n force, take this into account in the total energy. (Friction causes mechanical energy to be lost) K 1 +U 1 = K 2 +U 2 +E Heat K 1 +U 1 = K 2 +U 2 -W NC Physics 218, Lecture XII 65
Force and Potential Energy If we know the potential energy, U, we can find the force x du dx This makes sense For example, the force of gravity points down, but the potential increases as you go up Physics 218, Lecture XIII 66
Force and Potential Energy Draw some examples Gravity Spring Physics 218, Lecture XIII 67
Mechanical Energy We define the total mechanical energy in a system to be the kinetic energy plus the potential energy Define E K+U Physics 218, Lecture XIII 68
Conservation of Mechanical Energy For some types of problems, Mechanical Energy is conserved (more on this next week) E.g. Mechanical energy before you drop a brick is equal to the mechanical energy after you drop the brick K 2 +U 2 = K 1 +U 1 Conservation of Mechanical Energy E 2 =E 1 Physics 218, Lecture XIII 69
Friction and Springs A block of mass m is traveling on a rough surface. It reaches a spring (spring (p constant k) with speed V o and compresses it a total distance D. Determine t Physics 218, Lecture XV 70
Robot Arm A robot arm has a funny Force equation in 1- dimension 3x 2 F 1 X 0 x 2 0 F x0 where F 0 and X 0 are constants. The robot picks up a block at X=0 (at rest) and throws it, releasing it at X=X 0. What is the speed of the block? Physics 218, Lecture XV 71
Potential Energy Diagrams For Conservative forces can draw energy diagrams Equilibrium points Motion will move around the equilibrium If placed there with no energy, will just stay (no force) F x du dx 0 Physics 218, Lecture XV 72
A football is thrown A football of mass m starts at rest and is 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 Physics 218, Lecture XV 73
Problem 1 Physics 218, Chapter 7 & 8 74
Problem 1, continued Physics 218, Chapter 7 & 8 75
Problem 3 Physics 218, Chapter 7 & 8 76
Problem 3, continued Physics 218, Chapter 7 & 8 77
Unknown Force Physics 218, Chapter 7 & 8 78