Chapter 7 Potential energy and conservation of energy I Potential energy Energy of configuration II Work and potential energy III Conservative Non conservative forces IV Determining potential energy values Gravitational potential energy Elastic potential energy I V Conservation of mechanical energy VI External work and thermal energy VII External forces and internal energy changes VIII Power I Potential energy Energy associated with the arrangement of a system of objects that exert forces 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 energy If 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 s kinetic energy 1 U W Also valid for elastic potential energy Spring compression Spring force does W on block energy transfer from kinetic energy of the block to potential elastic energy of the spring fs Spring extension fs Spring force does W on block energy transfer from potential energy of 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 When system configuration changes force does work on the object W 1 transferring energy between KE of the object and some other form of energy of the system When the configuration change is reversed force reverses the energy transfer doing W 2 III Conservative Nonconservative forces If W 1 W 2 always conservative force Examples Gravitational force and spring force associated potential energies If W 1 W 2 nonconservative force Examples Drag force frictional force KE energy Non reversible process transferred into thermal Thermal energy Energy associated with the random movement of atoms and molecules This is not a potential energy 2 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 The net work it does on a particle moving between two points does not depend on the particle s path Conservative force W ab 1 W ab 2 Proof W ab 1 W ba 2 0 W ab 1 W ba 2 W ab 2 W ba 2 W ab 2 W ab 1 IV Determining potential energy values xf W x F x dx U Force F is conservative i Gravitational potential energy yf yi yf yi U mg dy mg y mg y f yi mg y U i 0 yi 0 U y mgy Change in the gravitational potential energy of the particle Earth system Reference configuration The 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 xf Elastic potential energy U kx dx x i k 2 x 2 xf xi 1 2 1 2 kx f kxi 2 2 Change in the elastic potential energy of the spring block system Reference configuration when the spring is at its relaxed length and the block is at xi 0 1 U i 0 xi 0 U x kx 2 2 Remember Potential energy is always associated with a system V Conservation of mechanical energy Mechanical energy of a system Sum of its potential U and kinetic K energies 3 Emec U K Assumptions Only conservative forces cause energy transfer within the system The system is isolated from its environment No external force from an object outside the system causes energy changes inside the system W K W U K U 0 K 2 K1 U 2 U1 0 K 2 U 2 K1 U1 Emec K U 0 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 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 y Emec constant x Emec K U 0 K 2 U 2 K1 U1 Potential energy curves Finding the force analytically U x W F x x F x dU x 1D motion dx 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 4 Turning point a point x at which the particle reverses its motion K 0 K always 0 K 0 5mv2 0 Examples x x1 Emec 5J 5J K K 0 x x1 Emec 5J 5J K K 0 impossible Equilibrium points where the slope of the U x curve is zero F x 0 U F x dx U dx F x U x dx F x Slope Equilibrium points Emec 1 Emec 2 Emec 3 Example x x5 Emec 1 4J 4J K K 0 and also F 0 x5 neutral equilibrium x2 x x1 x5 x x4 Emec 2 3J 3J K K 0 Turning points x3 K 0 F 0 particle stationary Unstable equilibrium x4 Emec 3 1J 1J K K 0 F 0 it cannot move to x x4 or x x4 since then K 0 Stable equilibrium 5 Review Potential energy W U The zero is arbitrary Only potential energy differences have physical meaning The potential energy is a scalar function of the position The force 1D is given by F dU dx P1 The force between two atoms in a diatomic molecule can be represented by the following potential energy function a 12 a 6 U x U 0 2 x x i Calculate the force Fx where U0 and a are constants F x a a 11 a a 5 dU x U 0 12 2 2 2 6 dx x x x x U 0 12a12 x 13 12a 6 x 7 12U 0 a a 13 a 7 x x ii Minimum value of U x U x min if x a 13 7 12U 0 a a dU x F x 0 0 dx a x x U a U 0 1 2 U 0 U0 is approx the energy necessary to dissociate the two atoms 6 VI Work done on a system by an external force Work is energy transfer to or from a system by means of an external force acting on that system When more than one force acts on a system their net work is the energy transferred to or from the system No Friction W Emec K U Ext force Emec K U 0 only when Remember System isolated No ext forces act on a system All internal forces are conservative Friction F f k ma v 2 v02 2ad a 0 5 v 2 v02 d m 2 2 1 1 1 v v0 Fd …