1Chapter 7 – Potential energy and conservation of energyI. Potential energy  Energy of configurationII. Work and potential energyIII. Conservative / Non-conservative forcesIV. Determining potential energy values:- Gravitational potential energy- Elastic potential energyI. V. Conservation of mechanical energyVI. External work and thermal energyVII. External forces and internal energy changesVIII. PowerI. Potential energyEnergy associated with the arrangement of a system of objects that exertforces on one another. Units: J Examples:- Gravitational potential energy: associated with the state of separation between objects which can attract one another via the gravitational force.- Elastic potential energy: associated with the state of compression/extension of an elastic object. II. Work and potential energyIf tomato rises  gravitational force transfers energy “from”tomato’s kinetic energy “to” the gravitational potential energy of the tomato-Earth system.If tomato falls down  gravitational force transfers energy “from” the gravitational potential energy “to” the tomato’skinetic energy.2WU−=∆Also valid for elastic potential energySpring force does –W on block  energy transfer from kinetic energy of the block to potential elastic energy of the spring.Spring force does +W on block energy transfer from potential energyof the spring to kinetic energy of the block.General:- System of two or more objects.- A force acts between a particle in the system and the rest of the system.fsfsSpring compressionSpring extension- When the configuration change is reversed  force reverses the energy transfer, doing W2.III. Conservative / Nonconservative forces- If W1=W2always  conservative force.Examples: Gravitational force and spring force  associated potentialenergies.- If W1≠W2 nonconservative force.Examples: Drag force, frictional force  KE transferred into thermal energy. Non-reversible process.- When system configuration changes  force does work on the object (W1) transferring energy between KE of the object and some other form of energy of the system.- Thermal energy: Energy associated with the random movement of atomsand molecules. This is not a potential energy.3- Conservative force: The net work it does on a particle moving around every closed path, from an initial point and then back to that point is zero.Conservative force  Wab,1= Wab,2Wab,1+ Wba,2=0  Wab,1= -Wba,2Wab,2= - Wba,2IV. Determining potential energy values∫∆−==fixxUdxxFW )(Force F is conservativeGravitational potential energy:Change in the gravitationalpotential energy of theparticle-Earth system.[]∫∆=−==−−=∆fifiyyifyyymgyymgymgdymgU )()(- The net work it does on a particle moving between two points does not depend on the particle’s path.Proof: Wab,2= Wab,1mgyyUyUii=→== )(0,0The gravitational potential energy associated with particle-Earth system depends only on particle’s vertical position “y” relative to the reference position y=0, not on the horizontal position.Reference configurationElastic potential energy:Change in the elastic potential energy of the spring-block system.[]22221212)(ixxfxxkxkxxkdxkxUfifi∫−==−−=∆221)(0,0 kxxUxUii=→==Reference configuration  when the spring is at its relaxed length and the block is at xi=0.Remember! Potential energy is always associated with a system.V. Conservation of mechanical energyMechanical energy of a system: Sum of its potential (U) and kinetic (K) energies.4Emec= U + KAssumptions: - Only conservative forces cause energy transfer withinthe system.UWKW∆−=∆=112212120)()(0 UKUKUUKKUK +=+→=−+−→=∆+∆- In an isolated system where only conservative forces cause energy changes, the kinetic energy and potential energy can change, but their sum, the mechanical energy of the system cannot change.∆Emec= ∆K + ∆U = 0- When the mechanical energy of a system is conserved, we can relate the sum of kinetic energy and potential energy at one instant to that at another instant without considering the intermediate motion and without finding the work done by the forces involved.-The system is isolated from its environment  No external force from an object outside the system causes energy changes inside the system.Emec= constant11220UKUKUKEmec+=+=∆+∆=∆Potential energy curvesFinding the force analytically:xy)1()()()()( motionDdxxdUxFxxFWxU −=→∆−=−=∆- The force is the negative of the slope of the curve U(x) versus x.-The particle’s kinetic energy is: K(x) = Emec– U(x)5Turning point: a point x at which the particle reverses its motion (K=0).Equilibrium points: where the slope of the U(x) curve is zero  F(x)=0∆U = -F(x) dx  ∆U/dx = -F(x)K always ≥0 (K=0.5mv2 ≥0 )Examples: x= x1Emec= 5J=5J+K  K=0x<x1 Emec= 5J= >5J+KK<0  impossibleExample: x ≥ x5 Emec,1= 4J=4J+K  K=0 and also F=0 x5neutral equilibriumEmec,1Emec,2Emec,3∆U(x)/dx = -F(x)  SlopeEquilibrium pointsx4 Emec,3=1J=1J+K  K=0, F=0, it cannot move to x>x4or x<x4, since then K<0 Stable equilibriumx2>x>x1, x5>x>x4 Emec,2= 3J= 3J+K  K=0  Turning pointsx3 K=0, F=0  particle stationary  Unstable equilibrium6Review: Potential energy W = -∆U-The zero is arbitrary  Only potential energy differences have physical meaning.- The force (1D) is given by: F = -dU/dx- The potential energy is a scalar function of the position.P1. The force between two atoms in a diatomic molecule can be represented by the following potential energy function:−=61202)(xaxaUxUwhere U0and a are constants.i) Calculate the force Fx[ ]−=+−−=−−−−=−=−−713076131205211201212126212)()(xaxaaUxaxaUxaxaxaxaUdxxdUxFii) Minimum value of U(x).[ ]007130min21)(0120)()()(UUaUaxxaxaaUxFdxxdUifxU−=−==→=−−→=−=U0is approx. the energy necessary to dissociate the two atoms.7VI. Work done on a system by an external forceNo Friction:dvvaadvvmafFk/)(5.02202202−=→+==−Work is energy transfer “to” or “from” a system by means