Potential Energy Curves

8/13/2019 Potential Energy Curves

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S O F A R W E H A V E D E A L T W I T H T W O

K I N D S O F P O T E N T I A L E N E R G Y :G R A V I T A T I O N A L ( U = M G H )

E L A S T I C ( U = 1 / 2 K X 2)

S O M E T I M E S I T I S M O R E H E L P F U L W H E NY O U H A V E P O T E N T I A L E N E R G Y A S A

F U N C T I O N O F P O S I T I O N T O F I N DP O T E N T I A L E N E R G Y F I R S T A N D D E R I V E

T H E F O R C E F R O M T H E R E .

Potential Energy Curves


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Force and Potential Energy

We established last class that the relationshipbetween work and potential energy was:

which leads toW U

F x U

UFx


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which leads to.

A conservative force always acts to push the systemtoward a lower potential energy.

dUF

dx

21

2

g

s

dF mgh mgdx

dF kx kx

dx


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A FORCE PARALLEL TO THE X -A XIS

ACTS ON A PARTICLE MOVING ALONGTHE X-AXIS. THE FORCE PRODUCESA POTENTIAL ENERGY: U(X) = 1 .2X 4.

WHAT IS THE FORCE WHEN THE

PARTICLE IS AT X=-0.8M?

Example


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This analysis can be extended to apply to threedimensions:

U U UF i j k

x y x


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A P A R T I C L E M O V I N G A L O N G T H E X - A X I S I S

A C T E D O N B Y A C O N S E R V A T I V E F O R C E . A T AC E R T A I N P O I N T , T H E F O R C E I S Z E R O .

W H A T D O E S T H I S T E L L Y O U A B O U T T H EV A L U E O F T H E P O T E N T I A L E N E R G Y

F U N C T I O N A T T H I S P O I N T ?

Check Point


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Potential Energy Curves for a Spring

Note: When the spring is either in a

state of maximum extension orcompression its potential energyis also a maximum

When the spring's displacementis DOWN the restoring force is

UP When the potential energy

function has a negative slope, therestoring force is positive and

vice-versa When the restoring force is zero,

the potential energy is zero At any point in the cycle, the total

energy is constant:U + K = Umax= Kmax


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Potential Energy Curve for a function


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Points of Equilibrium

When the force acting on the object is zero, the objectis said to be in a state of EQUILIBRIUM!

STABLE EQUILIBRIUM located at minimums, if

the object is displaced slightly it will tend backtothis location.

UNSTABLE EQUILIBRIUM located at maximums,if the object is displaced slightly it will tendaway

from this location. STATIC EQUILIBRIUM located at plateaus, where

the net force equals zero.


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Points of Equilibrium

Stable Equilibriumat x3and x5.

UnstableEquilibrium at x4.

Static Equilibriumat x1and x6.


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Turning Points

Define the boundariesof the particlesmotion.

We know that E=K+U,so where U=E, K=0Jand the particlechanges direction.

For instance, if E=4J,there would be turning

point at x2.

.


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If E = 1J, why isthe grey areareferred to as an

energy well?

Turning Points


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ExampleA particle of mass 0.5 kg obeys the

potential energy function:

U(x) = 2(x - 1) - (x - 2)3

What is the value of U(0)?

What are the values of x1and x2?


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U(x) = 2(x - 1) - (x - 2)3

How much potential energy does theparticle have at position x1?

If the object was initially releasedfrom rest, how fast is it moving as it

passes through position x1?


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U(x) = 2(x - 1) - (x - 2)3

How much potential energy doesthe mass have at x2?

How fast is it moving throughposition x2?


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U(x) = 2(x - 1) - (x - 2)3

Which position, x1or x2, is aposition of stable equilibrium?


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U(x) = 2(x - 1) - (x - 2)3

How fast is the particle movingwhen its potential energy, U(x) =0J?

If x3= x1, then how fast is theparticle moving as it passes throughposition x

3?


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U(x) = 2(x - 1) - (x - 2)3

Sketch the graph of the particle's

acceleration as a function of x.Indicate positions x1and x2onyour graph.


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ExampleA particle of mass 0.5 kg obeys the potential energy

function:U(x) = 2(x - 1) - (x - 2)3

At what value of x does the particle experience itgreatest negative acceleration?

What is the value of its potential energy at thisposition?

How much kinetic energy does it have at thisposition?

What force is being exerted upon it at thisposition?

What is the value of its acceleration at thisposition?

Potential Energy Curves

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