Conservation of Energy
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1 Lecture 3 Chapter 8 Physics I Conservation of Energy Course website: Lecture Capture: , Spring 04, Lecture 3
2 Outline Chapter 8 Conservative & Non-conservative forces Potential Energy Gravitational Pot. Energy Elastic Pot. Energy Conservation of Mechanical Energy 95.4, Spring 04, Lecture 3
3 Conservative Forces (definition) The work done by a conservative force in moving an object from point A to point B depends only on the positions A and B, not the path or the velocity of the object F B C A OR. The work done by a conservative force for a round trip and returning an object to its initial position is zero W A W B W Work done by F is the same for any path C F Conservative forces: gravity, spring Non-conservative forces: friction 95.4, Spring 04, Lecture 3
4 y y Gravitational force is conservative (proof) mg K d y K x Consider a block sliding down on a frictionless surface under the influence of gravity F G dl mg mg( ˆ) j dx( iˆ) dy( ˆ) j The work done by the gravitational force: W W G G F dl y G y mgdy mg( ˆ) j [ dx(ˆ) i mg( y y) dy( ˆ)] j The work done by gravity depends only on coordinates of the final and initial positions, so gravitational force is conservative WA W B A 95.4, Spring 04, Lecture 3 B
5 Conservation of Mechanical Energy!!! From the previous slide mv mgy mv mgy W G Work-Kinetic Energy Principle mg( y y) W G K mv mv mg( y y ) W G mv mv K combine it with U K U Let s introduce: Gravitational potential energy Total Mechanical Energy U mgy E K U (a new form of energy) E E so, we got 95.4, Spring 04, Lecture 3 E constant Which is Conservation of Mechanical Energy
6 Potential Energy (in general) For a system, where only conservative forces do work, we have: K U K U ( U U K K W ) Work-KE Principle U W K So, Relation between potential energy and work: U 95.4, Spring 04, Lecture 3 U W W F dl In general, we define the change in potential energy associated with a conservative force F as the negative of the work done by that force. Only changes of potential energy important, not absolute values Choose a suitable reference U=0 for each problem
7 Potential Energy Energy defined as the ability to do work Kinetic Energy: associated with energy of motion K mv Kinetic Energy Other types of stored energy that can do work A compressed spring An object at a height that can roll or drop These systems have the potential to do work Call it a stored potential energy Potential energy can only be associated with conservative forces 95.4, Spring 04, Lecture 3
8 Roller coaster The roller-coaster car starts from rest at the top of the hill. The height of the hill is 40 m. Calculate a) the speed of the car at the bottom of the hill; b) at what height it will have half this speed. 95.4, Spring 04, Lecture 3
9 Elastic/Spring Potential Energy Use a relation between potential energy and work: 0 U 95.4, Spring 04, Lecture 3 What is the potential energy of a spring compressed from equilibrium by a distance x? U ( x) U (0) U spring kx F x kx U W x F dl Choose U(0) = 0 (we have this freedom) x U ( x) kxdx kx 0 Potential energy of a spring 0 S
10 Brick/spring on a track A kg mass, with an initial velocity of 5 m/s, slides down the frictionless track shown below and into a spring with spring constant k=50 N/m. How far is the spring compressed? 95.4, Spring 04, Lecture 3
11 How to get a Force if a Potential Energy is given? Force Potential Energy Given a conservative force as a function of position, the change in potential energy associated with this (conservative) force is: U U(x)U(0) 0 0 x F x dx U( x) x 0 F x dx Potential Energy Force Given a potential energy as a function of position, the associated conservative force is: F(x) du(x) dx 95.4, Spring 04, Lecture 3
12 Example: Force Potential Energy Given the potential energy: U(x) Ax Bx C find the force F as a function of x F(x) du dx d dx (Ax Bx C) Ax B 95.4, Spring 04, Lecture 3
13 ConcepTest Paul and Kathleen start from rest at the same time on frictionless water slides with different shapes. At the bottom, whose velocity is greater? Water Slide I A) Paul B) Kathleen C) both the same Conservation of Energy: E i E f therefore: mgh mv v gh Because they both start from the same height, they have the same velocity at the bottom. Ref. level U=0
14 ConcepTest Paul and Kathleen start from rest at the same time on frictionless water slides with different shapes. Who makes it to the bottom first? Water Slide II A) Paul B) Kathleen C) both the same Even though they both have the same final velocity, Kathleen is at a lower height than Paul for most of her ride. Thus, she always has a larger velocity during her ride and therefore arrives earlier! Ref. level U=0
15 Example: Dropping ball An object of mass m is dropped v i 0 from a height h above the ground. Find speed of the object as it hits the ground: Kinematic equations From N. nd law we got this kinematic eq-n: 0 v f v i gh Energy conservation mv f K f U mgy f 0 f h Ref. level U=0 K mv mv f mgh i y U i i mgy v i 0 v f i? 0 h v f gh v f gh Thus, both approaches are equivalent 95.4, Spring 04, Lecture 3
16 Thank you See you on Wednesday 95.4, Spring 04, Lecture 3
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