Chapter 7 Potential Energy

Potential Energy Potential energy is the energy associated with the configuration of a system of objects that exert forces on each other The forces are internal to the system

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

System Example This system consists of Earth and a book Do work on the system by lifting the book through Dy The work done is mgyb - mgya

Potential Energy The energy storage mechanism is called potential energy A potential energy can only be associated with specific types of forces Potential energy is always associated with a system of two or more interacting objects

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 The work done on the object is done by and the upward displacement is

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

Gravitational Potential Energy, final The gravitational potential energy depends only on the vertical height of the object above Earth’s surface In solving problems, you must choose a reference configuration for which the gravitational potential energy is set equal to some reference value, normally zero The choice is arbitrary because you normally need the difference in potential energy, which is independent of the choice of reference configuration

Conservation of Mechanical Energy The mechanical energy of a system is the algebraic sum of the kinetic and potential energies in the system Emech = K + Ug The statement of Conservation of Mechanical Energy for an isolated system is Kf + Ugf = Ki+ Ugi An isolated system is one for which there are no energy transfers across the boundary

Conservation of Mechanical Energy, example Look at the work done by the book as it falls from some height to a lower height Won book = DKbook Also, W = mgyb – mgya So, DK = -DUg

Conservative Forces A conservative force is a force between members of a system that causes no transformation of mechanical 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

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

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

Elastic Potential Energy Elastic Potential Energy is associated with a spring, Us = 1/2 k x2 The work done by an external applied force on a spring-block system is W = 1/2 kxf2 – 1/2 kxi2 The work is equal to the difference between the initial and final values of an expression related to the configuration of the system

Elastic Potential Energy, cont This expression is the elastic potential energy: Us = 1/2 kx2 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

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 x2 will always be positive

Conservation of Energy, Extended Including all the types of energy discussed so far, Conservation of Energy can be expressed as DK + DU + DEint = DE system = 0 or K + U + E int = constant K would include all objects U would be all types of potential energy

Problem Solving Strategy – Conservation of Mechanical Energy Conceptualize: Define the isolated system and the initial and final configuration of the system The system may include two or more interacting particles The system may also include springs or other structures in which elastic potential energy can be stored Also include all components of the system that exert forces on each other

Problem-Solving Strategy, 2 Categorize: Determine if any energy transfers across the boundary If it does, use the nonisolated system model, DE system = ST If not, use the isolated system model, DEsystem = 0 Determine if any nonconservative forces are involved

Problem-Solving Strategy, 3 Analyze: Identify the configuration for zero potential energy Include both gravitational and elastic potential energies If more than one force is acting within the system, write an expression for the potential energy associated with each force

Problem-Solving Strategy, 4 If friction or air resistance is present, mechanical energy of the system is not conserved Use energy with non-conservative forces instead The difference between initial and final energies equals the energy transformed to or from internal energy by the nonconservative forces

Problem-Solving Strategy, 5 If the mechanical energy of the system is conserved, write the total energy as Ei = Ki + Ui for the initial configuration Ef = Kf + Uf for the final configuration Since mechanical energy is conserved, Ei = Ef and you can solve for the unknown quantity

Mechanical Energy and Nonconservative Forces In general, if friction is acting in a system: DEmech = DK + DU = -ƒkd DU is the change in all forms of potential energy If friction is zero, this equation becomes the same as Conservation of Mechanical Energy

Nonconservative Forces, Example 1 (Slide) DEmech = DK + DU DEmech =(Kf – Ki) + (Uf – Ui) DEmech = (Kf + Uf) – (Ki + Ui) DEmech = 1/2 mvf2 – mgh = -ƒkd

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 DEmech = -ƒkd

Conservative Forces and Potential Energy Define a potential energy function, U, such that the work done by a conservative force equals the decrease in the potential energy of the system The work done by such a force, F, is DU is negative when F and x are in the same direction

Conservative Forces and Potential Energy The conservative force is related to the potential energy function through The conservative force acting between parts of a system equals the negative of the derivative of the potential energy associated with that system This can be extended to three dimensions

Conservative Forces and Potential Energy – Check Look at the case of an object located some distance y above some reference point: This is the expression for the vertical component of the gravitational force

Nonisolated System in Steady State A system can be nonisolated with 0 = DT This occurs if the rate at which energy is entering the system is equal to the rate at which it is leaving There can be multiple competing transfers

Nonisolated System in Steady State, House Example

Potential Energy for Gravitational Forces Generalizing gravitational potential energy uses Newton’s Law of Universal Gravitation: The potential energy then is

Potential Energy for Gravitational Forces, Final The result for the earth-object system can be extended to any two objects: For three or more particles, this becomes

Electric Potential Energy Coulomb’s Law gives the electrostatic force between two particles This gives an electric potential energy function of

Energy Diagrams and Stable Equilibrium The x = 0 position is one of stable equilibrium Configurations of stable equilibrium correspond to those for which U(x) is a minimum x=xmax and x=-xmax are called the turning points

Energy Diagrams and Unstable Equilibrium Fx = 0 at x = 0, so the particle is in equilibrium For any other value of x, the particle moves away from the equilibrium position This is an example of unstable equilibrium Configurations of unstable equilibrium correspond to those for which U(x) is a maximum

Neutral Equilibrium Neutral equilibrium occurs in a configuration when U is constant over some region A small displacement from a position in this region will produce neither restoring nor disrupting forces

Potential Energy in Fuels Fuel represents a storage mechanism for potential energy Chemical reactions release energy that is used to operate the automobile