1 Chapter 7 Potential Energy

2 7.1 Potential Energy Potential energy is the energy associated with the configuration of a system of two or more interacting objects or particles that exert forces on each other The forces are internal to the system

3 Types of Potential Energy There are many forms of potential energy, including: Gravitational Electromagnetic Chemical Nuclear One form of energy in a system can be converted into another

4 System Example This system consists of the Earth and a book An external force does work on the system by lifting the book through  y The work done by the external force on the book is mgy b - mgy a Fig 7.1

5 Gravitational Potential Energy Gravitational Potential Energy is associated with an object at a given distance above Earth’s surface Assume the object is in equilibrium and moving at constant velocity upwardly. The external force is The displacement is

6 Gravitational Potential Energy, cont The quantity mgy is identified as the gravitational potential energy, U g U g = mgy Units are joules (J)

7 Gravitational Potential Energy, final The gravitational potential energy depends only on the vertical height of the object above Earth’s surface A reference configuration of the system must be chosen so that the gravitational potential energy at the reference configuration is set equal to zero The choice is arbitrary because the difference in potential energy is independent of the choice of reference configuration

8 Potential Energy Similar as the work-kinetic energy theorem, the work equals to the difference between the final and initial values of some quantity of the system. The quantity is called potential energy of the system. Through the work, energy is transferred into the system in a form different from kinetic energy. The transferred energy is stored in the system. The quantity U g = mgy is called the gravitational potential energy of the book-Earth system.

Conservation of Mechanical Energy for an isolated system, example The isolated system is the book and the Earth. The work done on the book by the gravitational force as it falls from some height y b to a lower height y a The motion of the book is free falling and the kinetic energy of the book increases. W on book = mgy b – mgy a = -  U g W on book =  K book = K a -K b (Work-kinetic theorem) So,  K = -  U g K b + U gb = K a + U ga Fig 7.2

10 Conservation of Mechanical Energy for isolated systems The mechanical energy of a system is the algebraic sum of the kinetic and potential energies in the system E mech = K + U g The Conservation of Mechanical Energy for an isolated system is K f + U gf = K i + U gi An isolated system is one for which there are no energy transfers across the boundary

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22 Elastic Potential Energy Elastic Potential Energy is associated with a spring, U s = 1/2 k x 2 The work done by an external applied force on a spring-block system is W = 1/2 kx f 2 – 1/2 kx i 2 The work is equal to the difference between the initial and final values of an expression related to the configuration of the system

23 Elastic Potential Energy, cont This expression is the elastic potential energy: U s = 1/2 kx 2 The elastic potential energy can be thought of as the energy stored in the deformed spring The stored potential energy can be converted into kinetic energy Fig 7.6

24 Elastic Potential Energy, final The elastic potential energy stored in a spring is zero whenever the spring is not deformed (U = 0 when x = 0) The energy is stored in the spring only when the spring is stretched or compressed The elastic potential energy is a maximum when the spring has reached its maximum extension or compression The elastic potential energy is always positive x 2 will always be positive

25 Conservation of Energy for isolated systems Including all the types of energy discussed so far, Conservation of Energy can be expressed as  K +  U +  E int =  E system = 0 or K + U + E int = constant K would include all objects U would be all types of potential energy The internal energy is the energy stored in a system besides the kinetic and potential energies.

Conservative Forces A conservative force is a force between members of a system that causes no transformation of mechanical energy to internal energy within the system The work done by a conservative force on a particle moving between any two points is independent of the path taken by the particle The work done by a conservative force on a particle moving through any closed path is zero

27 Nonconservative Forces A nonconservative force does not satisfy the conditions of conservative forces Nonconservative forces acting in a system cause a change in the mechanical energy of the system

28 Nonconservative Force, Example Friction is an example of a nonconservative force The work done depends on the path The red path will take more work than the blue path Fig 7.7

29 Mechanical Energy and Nonconservative Forces In general, if friction is acting in a system:  E mech =  K +  U = -ƒ k d  U is the change in all forms of potential energy If friction is zero, this equation becomes the same as Conservation of Mechanical Energy

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31 Fig 7.8

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36 Nonconservative Forces, Example 1 (Slide)  E mech =  K +  U  E mech =(K f – K i ) + (U f – U i )  E mech = (K f + U f ) – (K i + U i )  E mech = 1/2 mv f 2 – mgh = -ƒ k d

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41 Nonconservative Forces, Example 2 (Spring-Mass) Without friction, the energy continues to be transformed between kinetic and elastic potential energies and the total energy remains the same If friction is present, the energy decreases  E mech = -ƒ k d

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