Physics 218, Lecture XI1 Physics 218 Lecture 11 Dr. David Toback

Physics 218, Lecture XI2 Checklist for Today Things that were due Monday: –Chaps. 3 and 4 HW on WebCT –Progress on 5&6 problems Things that were due Tuesday: –Reading for Chapter 7 Things due for Wednesday’s Recitation: –Problems from Chap 5&6 Things due for Today: –Read Chapters 7, 8 & 9 Things due Monday –Chap 5&6 turned in on WebCT

Physics 218, Lecture XI3 Last time: Work This time More on Work Work and Energy –Work using the Work-Energy relationship Potential Energy Conservation of Mechanical Energy Chapters 7, 8 & 9 Cont

Physics 218, Lecture XI4

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6 Work in Two Dimensions You pull a crate of mass M a distance X along a horizontal floor with a constant 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, Lecture XI7 What if the Force is changing direction? What if the Force is changing magnitude?

Physics 218, Lecture XI8 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, Lecture XI9 Fancy sum notation  Integral 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.

Physics 218, Lecture XI10 Use an 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? Region of integration W=Fd

Physics 218, Lecture XI11 Non-Constant Force: Springs Springs are a good example of the types of problems we come back to over and over again! Hooke’s Law Force is NOT CONSTANT over a distance Some constant Displacement

Physics 218, Lecture XI12 Work done to stretch a Spring How much work do you do to stretch a spring (spring constant k), at constant velocity, from x=0 to x=D? D

Physics 218, Lecture XI13 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, Lecture XI14 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 statement: 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, Lecture XI15 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

Physics 218, Lecture XI16 Multiple ways to calculate the work done Multiple ways to calculate the velocity

Physics 218, Lecture XI17 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, Lecture XI18 Multiple ways to calculate velocity If we know the forces: If the force is constant –F=ma → V=V 0 +at, or V 2 -V 0 2 = 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 f 2 - ½mV i 2

Physics 218, Lecture XI19 Quick Problem I can do work on an object and it doesn’t change the kinetic energy. How? Example?

Physics 218, Lecture XI20 Problem Solving How do you solve Work and Energy problems? BEFORE and AFTER Diagrams

Physics 218, Lecture XI21 Problem Solving Before and After diagrams 1.What’s going 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, Lecture XI22 Before…

Physics 218, Lecture XI23 After…

Physics 218, Lecture XI24 Compressing a Spring A horizontal spring has spring constant 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, Lecture XI25 Potential Energy Things with potential: COULD do work –“This woman has great potential as an engineer!” Here we kinda mean the same thing E.g. Gravitation potential energy: –If you lift up a brick it has the potential to do damage

Physics 218, Lecture XI26 Example: Gravity & Potential Energy You lift up a brick (at rest) from the ground and then hold it at a height Z 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=mgZ –Gravitational potential energy

Physics 218, Lecture XI27 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 XI28 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 XI29 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, Lecture XI30 Problem Solving For Conservation of Energy problems: BEFORE and AFTER diagrams

Physics 218, Lecture XI31 Quick Problem 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, Lecture XI32 Next Week Reading for Next Time: –Finish Chapters 7, 8 and 9 if you haven’t already –Non-conservative forces & Energy Chapter 5&6 Due Monday on WebCT Start working on Chapter 7 for recitation next week

Physics 218, Lecture XI33

Physics 218, Lecture XI34 Compressing a Spring A horizontal spring has spring constant 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? 3.What is the speed if there is friction with coefficient  ?

Physics 218, Lecture XI35 Roller Coaster A Roller Coaster of mass M=1000kg starts at point A. We set Y(A)=0. What is the potential energy at height A, U(A)? What about at B and C? What is the change in potential energy as we go from B to C? If we set Y(C)=0, then what is the potential energy at A, B and C? Change from B to C

Physics 218, Lecture XI36 Kinetic Energy Take a body at rest, with mass m, accelerate for a while (say with constant force over a distance d). Do W=Fd=mad: V 2- V 0 2 = 2ad= V 2  ad = ½V 2 W = F. d = (ma). d= mad  ad = ½V 2  mad = ½ mV 2  W = mad = ½ mV 2 Kinetic Energy = ½ mV 2

Physics 218, Lecture XI37 Work and Kinetic Energy If V 0 not equal to 0 then V 2 - V 0 2 = 2ad W=F. d = mad = ½m (V 2 - V 0 2 ) = ½mV 2 - ½mV 0 2 =  (Kinetic Energy) W=  KE Net Work on an object (All forces)

Physics 218, Lecture XI38 A football is thrown A 145g football starts at rest and is thrown with a speed of 25m/s. 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 XI39 Example: Gravity Work by Gravity

Physics 218, Lecture XI40 Potential Energy in General Is the potential energy always equal to the work done on the object? –No, non-conservative forces –Other cases? What about for conservative forces?

Physics 218, Lecture XI41 Water Slide Who hits the bottom with a faster speed?

Physics 218, Lecture XI42 Mechanical Energy Consider a Conservative System W net =  K (work done ON an object)  U Total = -W net Combine  K = W net = -  U Total =>  K +  U = 0 Conservation of Energy

Physics 218, Lecture XI43 Conservation of Energy Define E=K+U  K +  U = 0 => (K 2 -K 1 ) +(U 2 -U 1 )=0 K 2 +U 2 = K 1 +U 1 Conservation of Mechanical Energy E 2 =E 1

Physics 218, Lecture XI44 Conservative vs. Non-Conservative Forces Nature likes to “conserve” certain types of things Keep them the same Kinda like conservative politicians Conservationists

Physics 218, Lecture XI45 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, Lecture XI46 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, Lecture XI47 Is Friction a Conservative Force?

Physics 218, Lecture XI48 Integral Examples we know…

Physics 218, Lecture XI49 Work done to stretch a Spring

Physics 218, Lecture XI50 Robot Arm A robot arm has a funny Force equation in 1-dimension where F 0 and X 0 are constants. What is the work done to move a block from position X 1 to position X 2 ?

Physics 218, Lecture XI51 Stretch a Spring A person pulls on a spring and stretches it a from the equilibrium point for a total distance D. At this distance the force required to keep the spring stretched is F. How much work is done on the spring in terms of the given variables? Can we use F. d?