Chapter 8

Potential Enegy

Introduction

• Potential Energy- Energy associated with the configuration of a system of objects that exert forces on each other. – Associated with conservative forces.– No Energy is lost/gained across the system

boundaries– Any gain in potential energy is coupled with a loss

of KE, and vice versa.– Conservation of Mechanical Energy

8.1 Potential Energy of a System

• Look at a system of multiple particles that interact with internal forces only. – The KE of the system is the sum of KE of each of

the particles.• In some cases the KE of an object within the system is

negligible, and can be ignored. (Ex: object falling to earth)

– Potential Energy is often considered an Energy storage mechanism

8.1

• Book/Earth System• Energy can be added to the system by an external force by lifting the object to a new height. (work is done)

8.1

• No Change in KE, no change in Temp (Eint)• Where did the energy go?• It must be stored somewhere.• If we let the book go and it returns to height

A…The book now has KE at height A that it didn’t have before.

8.1

• The work done on the system to lift the object gave it the “Potential” to have kinetic energy.

• Gravitational Potential Energy (Ug)rF appW

]ˆ)[()ˆ( jj ab yymgW

ab mgymgyW mgyU g

gUW

8.1

• Note that the work done is the same between lifting the object/pushing it up a ramp.

(Dot product of lift force and displacement are only share j components)

• Ug also depends on a reference “zero” height– This can be chosen at any level– Above is positive Ug, below is negative Ug

8.2 Isolated Systems (Cons. of ME)

• The mechanical energy of a system is the sum of K and U (kinetic and stored potential)

• The conservation of mechanical energy of a systems means the total MEinitial equals total MEfinal (assuming isolated, no outside work)

UKEmech

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8.2

• Quick Quiz p. 220

8.2

• Elastic Potential Energy – energy stored in a stretched compressed spring.– Work done on spring = energy stored in it.

– Energies for a spring system in Horizontal oscillation• At max compression, K = 0 J, Us = Etot

• At equilibrium, K = Etot, Us = 0 J

221 kxUW s

8.2

• Conservation of Mech Energy

sfgffsigii UUKUUK

0 sg UUK

8.2

• Quick Quiz p. 223• Example Problems 8.2-8.5

8.3 Conservative and Non-Conservative Forces

• As an object moves down towards earth (near the surface) the work done by gravity is the same regardless of the path taken.– Falling vs. Sliding down an incline

• Conservative Force- A force whose work is independent of the path taken (Gravity)– Also a conservative force does zero work for a

closed path (same starting ending points).

8.3

• Spring Force – Conservative• Non-Conservative Force- a force whose work

depends on the path taken. (Friction)– Conservative forces generally convert energy

between potential and kinetic– Non-Conservative forces generally convert energy

into a non-mechanical (non-recoverable form).

8.4 Changes in ME with Non-Conservative Forces

• For a situation with a non conservative force (usually friction)-

• Nonconservative forces reduce the amount of Mechanical Energy available as K or U.

Quick Quizzes p 230Examples 8.6-8.10

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8.5 Conservative Forces and Potential Energy

• Remember the work done by a conservative force is independent of path, only depends on initial/final configuration of the system.

• Potential Energy Function U such that the work done by a conservative force equals the decrease in potential energy of the system.

8.5

• Work by a conservative force(remember the conservative force varies with postion)

Rearranged…

And over tiny displacements…

f

i

x

xc UFxdxW

f

i

x

x if UFxdxxU )(

FxdxdU

8.5

• Therefore, the conservative force is related to the potential energy function through…

• That is, the x component of a conservative force acting on an object within a system equals the negative derivative of the potential energy of the systems with respect to x.

dx

dUFx

8.5

• Check with Us

• Check with Ug

Quick Quiz 8.11

8.6

• Energy Graphs (U vs. x) can be used to qualitatively interpret the motion of a system.

• Consider a block spring system-

8.6

• For any slight displacement from x = 0 the block will accelerate back to equilibrium.

• Stable Equilibrium• The natural tendencyis to get to the lowest state of potential energypossible.

8.6

• Consider another scenario where an object is at equilibrium, but has Umax. (Unstable Eq)

• A ball balanced on top of a hill• If it is displaced to eitherside of eq, the slope will allow the object to reduceits potential energy.

6.5

• Neutral Equilibrium- situation where a displacement does not allow for a decrease in potential energy (no change)

• A ball resting on a horizontal surface.• The Potential energy function is a constant

value.

Example 8.11