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Chapter 7

Potential Energy

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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

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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

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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 mgyb - mgya

Fig 7.1

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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

g

mF j)( ab yyr

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Gravitational Potential Energy, cont

The quantity mgy is identified as the gravitational potential energy, Ug

Ug = mgy Units are joules (J)

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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

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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 Ug=mgy is called the gravitational

potential energy of the book-Earth system.

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7. 2 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 yb to a lower height ya

The motion of the book is free falling and the kinetic energy of the book increases.

Won book = mgyb – mgya = -Ug

Won book = Kbook = Ka-Kb (Work-kinetic theorem)

So, K = -Ug Kb + Ugb = Ka + Uga Fig 7.2

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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 Emech = K + Ug

The 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

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Fig 7.4

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Fig 7.5

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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 kxf

2 – 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

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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

Fig 7.6

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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

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Conservation of Energy for isolated systems Including all the types of energy discussed so

far, Conservation of Energy can be expressed as

K + U + Eint = 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.

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7.3 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

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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

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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

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Mechanical Energy and Nonconservative Forces In general, if friction is acting in a

system: Emech = K + U = -ƒkd 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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Fig 7.8

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Nonconservative Forces, Example 1 (Slide)

Emech = K + U

Emech =(Kf – Ki) +

(Uf – Ui)

Emech = (Kf + Uf) –

(Ki + Ui)

Emech = 1/2 mvf2 –

mgh = -ƒkd

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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

Emech = -ƒkd

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