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Chapter 7

Potential Energyand

Conservation of Energy

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Forms of EnergyThere are many forms of energy, but they can all be put into two categories

KineticKinetic energy is energy of motionPotentialPotential energy is energy of position

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Kinetic Energy is energy of motion

Electrical Energy is the movement of electrical charges.Thermal Energy, or heat, is the internal energy in substances––the vibration and movement of the atoms and molecules within substances. Radiant Energy is electromagnetic energy that travels in transverse waves.Motion Energy is the movement of objects and substances from one place to another. Sound is the movement of energy through substances in longitudinal (compression/rarefaction) waves

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Potential energy is energy of position

Gravitational Energy is the energy of position or place. Chemical Energy is energy stored in the bonds of atoms and molecules. Nuclear Energy is energy stored in the nucleus of an atom––the energy that holds the nucleus together. Stored Mechanical Energy is energy stored in objects by the application of a force.

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Part 1

Potential Energy

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Potential Energy is energy of positionPotential energy U is energy that can be associated with the configuration of a system of objects that exert forces on one another.

If the configuration of the system changes, then the potential energy of the system can also change

Types of potential energy studied in University Physics courses

Gravitational Potential Energy Elastic Potential Energy

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Work and Potential EnergyChange in potential energy

WUUU if −=−=Δ

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Determining Potential Energy Values1D case

3D case

Attention: the equation can NOT be used for frictional forces (see later “conservative and non-conservative forces”)

∫−=−=Δf

i

x

xif dxxFUUU )(

∫ ⋅−=−=Δf

i

r

rif rdrFUUU

r

r)(

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Gravitational Potential EnergyGravitational potential energy is associated with the state of separation between objects with masses

)()()( if

y

y

f

i

yymgdymgdxxFUf

i

−=−−=−=Δ ∫∫

ymgU Δ=Δ

Only changes in potential energy are physically meaningful. However to simplify calculations we may select a reference point yi where Ui=0, then

mgyyU =)(

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Note:The gravitational potential energy associated with a particle-Earth system depends ONLY on the vertical position y (or height) of the particle relative to the reference position (y=0), not on the horizontal position

mgyU =

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Elastic Potential EnergyGravitational potential energy is associated with the state of compression or extension of an elastic (spring-like) objects. A good approximation for many springs is Hook’s law

thenkxF −=

2

21)( kxxU =

)(21)()( 22

if

x

x

f

i

xxkdykxdxxFUf

i

−=−−=−=Δ ∫∫Choosing the reference point when the spring is relaxed

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Conservative forces or more mathematicsGeneral definition of conservative force:

Force is conservative if work that it does around a closed curve is zero. Equivalent to statement: for conservative force, work is independent of the path that connects initial and final points

There is a potential energy associated with a conservative force

Since, frictional force does negative work in both directions, there is no potential energy associated with the frictional force

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Path Independence

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Introducing potential

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Potential and force

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One-dimensional case

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“Zero” of Potential Energy

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Constant gravitational force

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Elastic spring

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Part 2

Conservation of Mechanical Energy

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Conservation of Mechanical EnergyThe mechanical energy of a system is the sum of its potential energy U and the kinetic energy K of the objects within it

UKE +=mech

In an isolated system where only conservative forces cause energy change, the kinetic energy and potential energy can change, but their sum, the mechanical energy of the system cannot change

ffii UKUK +=+

22

little math

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on practical sideWhen the mechanical energy of a system is conserved (the system is isolated and forces are conservative), we can relate the sum of kinetic energies and potential energies at one instant to that at another instant without considering the intermediate motion

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Finding the force from potential (in 1D)Potential U(x)

Force F(x) dxxdUxF )()( −=

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Energy diagrams

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Turning points

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Equilibrium points

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Bungee jumpStudent jumps off a bridge 52 meters above a river with a bungee cord tied around his ankle. He falls 15 meters before the bungee cord begins to stretch. Student’s mass is 75 kg and the cord (spring) constant is k=50 N/m.

If we neglect air resistance, estimate how far below the bridge the student would fall before coming to stop.

example

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solutionexample

ffii UKUK +=+

021

21

)(

2

2

=−−

=+

+=+

mgLmgdkd

kdUK

dLmgUK

ff

ii

for k=50 N/m, L=15 m, m=75 kg

d=40 m and L+d = 55 m

the bridge is 52 meters … oops

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Part 3

Conservation of Energy

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Conservation of EnergyThe total energy of a system can change only by amounts of energy that are transferred to or from that system

internalthermal EEUKE Δ+Δ+Δ+Δ=Δ

The total energy of an isolated system cannot change

0internalthermal =Δ+Δ+Δ+Δ EEUK

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Conservation of Energy (more)The law of conservation of energy says that energy is neither created nor destroyed. When we use energy, it doesn’t disappear. We change it from one form of energy into another.

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Energy EfficiencyEnergy efficiency is the amount of useful energy you get from a system. A perfect, energy-efficient machine would change all the energy put in it into useful.

Converting one form of energy into another form always involves a loss of usable energy.

Example: human body is a very inefficient “machine”. Fuel is food. (gives the energy to move, breathe, think). Human body is less than five percent efficient most of the time. The rest of the energy is lost as heat.

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Sources of energyEnergy sources are classified into two groups—renewable and nonrenewable.

Energy can be converted into secondary energy sources like electricity and hydrogen.

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Part 4

Friction involved

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Simultaneous Presence of Conservative and Non-Conservative Forces

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Friction involvedFor physics 231 we may consider friction as a transfer to thermal energy.

for a constant frictional force

∫=f

i

x

x k dxxfE )(thermal

0thermal =Δ+Δ+Δ EUK

)( ifkffii xxfUKUK −++=+

)(thermal ifk xxfE −=

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Block on inclineexample

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Solutionexample