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Physics 218 Lecture 13 Dr. David Toback Physics 218, Lecture XIII 1 Checklist for Today •Things due for Last Thursday: – Read Chapters 7, 8 & 9 •Things that were due Last Monday: – Chap 5&6 turned in on WebCT •Things that were due for Wednesday’s Recitation: – Problems from Chap 7 •Things due for this coming Monday: – Problems from Chap 7 on WebCT – Chaps 5&6 if you haven’t done them Physics 218, Lecture XIII 2 already The Schedule This week: (2/25) • HW on Chaps 5&6 on WebCT • 3rd and 4th lectures (of six) on Chapters 7, 8 & 9 • Chapter 7 in recitation Next week: (3/3) • Chapter 7 due in WebCT • 5th and 6th lectures (of six) on Chapters 7, 8 & 9 • Chapter 8 in recitation Following week: (3/10) Spring Break!!! Following Week: (3/17) • Chapter 8 due in WebCT • Reading for Chapters 10 & 11 • Lecture on Chapters 10 & 11 • Chapter 9 and Exam 2 Review in recitation Following Week: (3/24) • Chapter 9 due in WebCT • Exam 2 on Tuesday • Recitation on Chapters 10 & 11 • Reading for Chapters 12 & 13 for Thursday • Lecture 12 & 13 on Thursday Physics 218, Lecture XIII 3 Chapters 7, 8 & 9 Cont Last time: – Work and Energy – The Work-Energy relationship This time and next time: – Potential Energy – Conservation of Mechanical Energy – Conservation of Energy – Lots of problems Physics 218, Lecture XIII 4 Physics 218, Lecture XIII 5 Different Style Than the Textbook I like teaching this material using a different style than the textbook 1. Teach you the concepts 2. Give you the important equations 3. Then we’ll do lots of problems Physics 218, Lecture XIII 6 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 XIII 7 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 XIII 8 Other Potential Energies: Springs Last week we calculated that it took ½kx2 of work to compress a spring by a distance x How much potential energy does it now how have? 2 U(x) = ½kx Physics 218, Lecture XIII 9 Force and Potential Energy If we know the potential energy, U, we can find the force Fx 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 10 Force and Potential Energy Draw some examples… –Gravity –Spring Physics 218, Lecture XIII 11 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 12 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 K2+U2 = K1+U1 Conservation of Mechanical Energy E2=E1 Physics 218, Lecture XIII 13 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 = 2 ½mv Physics 218, Lecture XIII + mgy 14 Problem Solving For Conservation of Energy problems: BEFORE and AFTER diagrams Physics 218, Lecture XIII 15 Conservation of Energy Problems Before… Physics 218, Lecture XIII 16 After Physics 218, Lecture XIII 17 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 XIII 18 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? • DU = U2-U1 = Wext = mg (h2-h1) • Write U = mgh • U=mgh + Const Only change in potential energy is really meaningful Physics 218, Lecture XIII 19 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. Before After Z Z C What is the spring constant? Physics 218, Lecture XIII 20 Quick Problem A refrigerator with mass M and speed V0 is sliding on a dirty floor with coefficient of friction m. Is mechanical energy conserved? Physics 218, Lecture XIII 21 Non-Conservative Forces • We’ve talked about three different types of forces: 1.Gravity: Conserves mechanical energy 2.Normal Force: Conserves mechanical energy (doesn’t do work) 3.Friction: Doesn’t conserve mechanical energy • Since Friction causes us to lose mechanical energy (doesn’t conserve mechanical Physics 218, Lecture XIII energy) it is a Non-Conservative force! 22 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! Physics 218, Lecture XIII 23 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 – 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 XIII 24 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 XIII 25 Roller Coaster with Friction A roller coaster of mass m starts at rest at height y1 and falls down the path with friction, then back up until it hits height y2 (y1 > y2). 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 XIII 26 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.) Wnet = 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) DUTotal = WPerson =-WGravity Physics 218, Lecture XIII 27 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. EHeat+Light+Sound.. = -WNC If work is done by a non-conservative force, take this into account in the total energy. (Friction causes mechanical energy to be lost) K1+U1 = K2+U2+EHeat… K1+U1 = K2+U2-WNC Physics 218, Lecture XIII 28 Friction and Springs A block of mass m is traveling on a rough surface. It reaches a spring (spring constant k) with speed Vo and compresses it a total distance D. Determine m Physics 218, Lecture XIII 29 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? Physics 218, Lecture XIII l l 30 Coming up… • Lectures: – Last lectures on Chaps 7, 8 and 9 • HW due in WebCT on Monday – Chapter 7 • Reading for Lecture next week – Chaps 10 & 11: Momentum • Recitation next week – Chapter 8 Physics 218, Lecture XIII 31 Physics 218, Lecture XIII 32 Roller Coaster You are in a roller coaster car of mass M that starts at the top, height Z, with an initial speed V0=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, Lecture XIII 33 Energy • Potential Energy & Conservation of Energy problems • The relationship between potential energy and Force • Energy diagrams and Equilibrium Physics 218, Lecture XIII 34 Energy Review 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.) Wnet = 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) DUTotal = WPerson =-WGravity Physics 218, Lecture XIII 35 Energy Review If work is done by a non-conservative force it is negative work (slows something down), and we get heat, light, sound etc. EHeat+Light+Sound.. = -WNC If work is done by a non-conservative force, take this into account in the total energy. (Friction causes mechanical energy to be lost) K1+U1 = K2+U2+EHeat… K1+U1 = K2+U2-WNC Physics 218, Lecture XIII 36 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) Fx Physics 218, Lecture XIII dU dx 0 37 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? Physics 218, Lecture XIII 38 Roller Coaster with Friction A roller coaster car of mass m starts at rest at height y1 and falls down the path with friction, then back up until it hits height y2 (y1 > y2). 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 XIII 39 Roller Coaster with Friction A roller coaster car of mass m starts at rest at height y1 and falls down the path with friction, then back up until it hits height y2 (y1 > y2). An odometer tells us that the total scalar distance traveled is d. Assuming we don’t know anything about the friction or the path, how much work is done by friction on this path? Assuming that the magnitude and angle of the force of friction, F, between the car and the track is constant, find |F|. Physics 218, Lecture XIII 40 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). How far does the cord stretch Dy? Physics 218, Lecture XIII l 41 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 XIII 42 Robot Arm A robot arm has a funny Force equation in 1dimension 2 3x F(x) F0 1 2 x where F0 and X0 are 0 constants. What is the work done to move a block from position X1 to position X2? Physics 218, Lecture XIII 43