Lectures 11-13: From Work to Energy Energy Conservation

Physics 218: sect.513-517 Lectures 11-13: From Work to Energy Energy Conservation Prof. Ricardo Eusebi for Igor Roshchin 1

Physics 218: sect.513-517 Energy and Work-Energy relationship Prof. Ricardo Eusebi for Igor Roshchin 2

The Schedule Week of February 20: Chapter 4 HW due in MP Chapters 6&7 in lecture (Q6.22-6.25, Q7.2-7.3) Chapter 5 in recitation Week of February 27: Chapter 5 HW due in MP Chapter 7 (and 6) continued in lecture (Q7.14-7.24) Chapter 6 +7(1,2) in recitation Week of March 5: Chapter 6 + Chapter 7a HW due in MP Chapter 8 in lecture (reading questions, Q8.6, Q8.13) Chapter 7 in recitation Week of March 19: Chapter 7b + 8a HW due in MP Exam 2 (!) March 20 Chapter 8 in Recitation Chapter 9 in lecture (reading questions due) 3

Example in Nature: Springs Springs are a good example of the types of problems we come back to over and over again! Hooke s Law F kx Some constant Displacement Force is NOT CONSTANT over a distance 4

Work done to stretch a Spring How much work do you do to stretch a spring (very slowly) from D x=0 to x=d? 5

Does the Earth do work on the Moon? 6

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 has energy now We say this box has kinetic energy! Think of it as Mechanical Energy or the Energy of Motion Kinetic Energy = ½mv 2 7

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 8

Summary of equations Kinetic Energy = ½mV 2 W= DKE Can use change in speed to calculate the work, or the work to calculate the speed 9

Multiple ways to calculate the work done Multiple ways to calculate the velocity 10

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 11

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= DKE = KE f -KE i = ½mV f 2 - ½mV i 2 12

Quick Problem I can do work on an object and it doesn t change the kinetic energy. How? Example? 13

Problem Solving How do you solve Work and Energy problems? BEFORE and AFTER Diagrams 14

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 15

Potential Energy Things with potential: COULD do work This woman has a great potential as an engineer! Here we mean the same thing E.g. Gravitation potential energy: If you lift up a brick it has the potential to do damage 16

Example: Gravity & Potential Energy You (very slowly) 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 17

Potential Energy The whole concept of Potential Energy is really just to simplify our life in standard cases (gravity, springs etc.) Rather than explicitly calculating work, just shift it into the definition of potential energy when you can No need to calculate work by these standard forces every time: Just use formula to calculate change in potential energy (e.g. mgh 2 -mgh 1 ) Change in P.E. is equal to the work W you were after Then use DKE=-DPE to get the velocity 18

Other Example: Springs Springs are a good example of the types of problems we come back to over and over again! F Hooke s Law kx Some constant Displacement Force is NOT CONSTANT over a distance 19

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? F fr F s mg F s 20

Mechanical Energy We define the total mechanical energy in a system to be the kinetic energy plus the potential energy Define E K+U 21

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 22

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 Gravity: E = K + U = ½mv 2 + mgy Spring: E = K + U = ½mv 2 + ½kx 2 23

Problem Solving For Conservation of Energy problems: BEFORE and AFTER diagrams 24

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? 25

This and Next Week Next Lecture: More on Work and Energy Finish the reading for Chapter 7 HW5 due next Monday Start working on HW 6 and 7a Recitation on Chapter 6 and 7(1-2) problems 26

Physics 218: sect.513-517 Conservative and Non-Conservative Forces Prof. Ricardo Eusebi for Igor Roshchin 28

Problem Solving For Conservation of Energy problems: BEFORE and AFTER diagrams 30

Before 31

After 32

Quick Problem A refrigerator with mass M and speed V 0 is sliding on a dirty floor with coefficient of friction m. Is mechanical energy conserved? 33

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! 34

Law of Conservation of Energy Mechanical Energy: NOT always conserved If 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 is conserved! 35

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 energy and kinetic energy keep turning into one another Is a uniquely defined as a function of the position (does not depend on other forces, acceleration, speed, etc.). 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) 36

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 37

Roller Coaster You are in a roller coaster car of mass M that starts at the top, height 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 38

Roller Coaster with Friction A roller coaster of mass m starts at rest at height y 1 and falls down the path with friction, then back up until it hits height y 2 (y 1 > y 2 ). An odometer tells us that the total scalar distance traveled is d. How much work is done by friction on this path? 39

Energy Summary If there is net work done 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 = DK If there is a change in the potential energy, some one had to do some work: (Ball falling 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 the ball) DU Total = W Person =-W Gravity 40

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 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 41

Force and Potential Energy If we know the potential energy, we can find the force F du x dx This makes sense For example, the force of gravity points down, but the potential increases as you go up 42

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 43

Mechanical Energy We define the total mechanical energy in a system to be the kinetic energy plus the potential energy Define E K+U 45

Problem Solving For Conservation of Energy problems: BEFORE and AFTER diagrams 48

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? 49

Energy Power Conservation of Mechanical Energy problems Conservative Forces Conservation of Energy 50

P av =DW/Dt Power P=lim DW/Dt= dw/dt (Dt ->0) Power is the time rate at which work is done. 1 W = 1 J/s 1 hp = 746 W (That s a VERY powerful horse!) 51

Power P av =DW/Dt = F Ds / Dt = F v av P=lim F Ds / Dt = F ds/dt = F v (Dt ->0) 52

Physics 218: sect.513-517 Potential Energy Stable and unstable equilibrium Prof. Ricardo Eusebi for Igor Roshchin 54

Potential Energy - Gravity 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? DU = 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 56

Potential Energy: 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 have now? U(x) = ½kx 2 57

Problem Solving For Conservation of Energy problems: BEFORE and AFTER diagrams 58

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. We observe that the spring compresses by distance C. What is the spring constant? Before Z After Z C 59

Stable vs. Unstable Equilibrium Points The force is zero at both maxima and minima but If I put a ball with no velocity there would it stay? What if it had a little bit of velocity? 63

Friction and Springs A block of mass m is traveling on a rough surface. It reaches a spring (spring constant k) with speed v o and compresses it by an amount D. Determine m mg 64

Problem m d h m 1 A block of mass m is at rest at the top, then travels along an inclined (angle q) rough surface (kin. fr. m 1 ), then a level rough surface. A spring (constant k) is at distance d from the incline. What should be m 2 for the box not to reach the spring? If it is known that the box reaches the spring and bounces back before stopping on the level surface, how much the spring compresses (length)? How far from incline it stops? m 2 65

Bungee Jump You are standing on a platform high in the air with a bungee cord (spring constant k) strapped to your leg. You have mass m and jump off the platform. 1. How far does the cord stretch, l in the picture? 2. What is the equilibrium point around which you will bounce? l l 66

Coming up Next time: Chapter 8 Exam 2 around the corner Homework 6b, 7a due on Monday 68