Consider an ideal spring. Elastic Potential Energy Hooke’s Law.

Consider an ideal spring.

Elastic Potential Energy

xkFs

NFs

mx

xF

E

s

e

vs. of graph a of

curve the under area the of negative the - Energy PotentialElastic

sF

x

2

2

1kxEe

Hooke’s Law

Gravitational Potential Energy

m

AF

Initial 0iym

AF

Final 0fy

A

f

F

y

m

force applied constant aby

velocity constant a at height a to

ground the from lifted is mass ofbox A

gF

AF

gA FF ?gF

constant,gF

negative constant,gF

(height) vs.

of graph the of curve the under area

the of negative the - Energy Potential nalGravitatio

yF

E

g

g

mgyEg

NFg

myy

gF

The Potential Energy Function & Equilibrium

xFU

of graph a of curve the under area the of negative The - Energy Potential

FdxU For one dimensional, conservative forces

Consider the force of gravity: mgFg

dymgU g

mgyEU gg


xFU

of graph a of curve the under area the of negative The - Energy Potential

FdxU For one dimensional, conservative forces

Consider the spring: Law sHooke' xkFs

dxkxU s 2

2

1kxEU es

JU

mx


xkFs

But

2

2

1kxEU es

JU

mx

What does the slope of this graph represent?

dx

dU slope

2

2

1kx

dx

d

dx

dU

kxdx

dU

Fdx

dU

graph the of slope xUdx

dUF

The Potential Energy Function & Equilibrium JU

mx


dUF

0F whenOccurs - mEquilibriu

negative. is force the so

positive, is slope the that Notice

positive. is force the so

negative, is slope the that Notice

Stable Equilibrium – occurs when a small displacement results in a restoring force that accelerates the particle back

towards the equilibrium position.


JU

mx


dUF

positive. is force the so

negative, is slope the that Notice

negative. is force the so

positive, is slope the that Notice

Unstable Equilibrium – occurs when a small displacement results in a force that accelerates the particle away from the equilibrium position.

Consider the potential energy function for a skier on the top of hill between two valleys.


JU

mx


dUF

Neutral Equilibrium – occurs when a small displacement results in zero force so that the particle is again in

equilibrium.

Consider the potential energy function for an object on a plateau.


Example An object has the potential energy function as given below.

20

-2-4-6-8

-10

JU

mx

3210

-1-2-3

NF

mx

2 4 121086 14

2 4 121086 14


dUF

Nm

J

24

8

Nm

J

14

4

a. Sketch the corresponding Force vs. Position graph.



20

-2-4-6-8

-10

JU

mx

3210

-1-2-3

NF

mx

2 4 121086 14

2 4 121086 14

b. Identify all points of equilibrium.

mx 5Stable Equilibrium

mxm 1411 Neutral Equilibrium



20

-2-4-6-8

-10

JU

mx

2 4 121086 14

graph. the on line double theby

shown as ofenergy mechanical total a has object the Suppose

xU

J5

? and atenergy kinetic sobject' the is Whatc. mxmx 94

totalE

kT EUE Total Mechanical Energy

xUEE Tk

xU

of graph the From

JmU 84 JmU 69

JJEk 85 JJEk 65

JEk 3 JEk 1



20

-2-4-6-8

-10

JU

mx

2 4 121086 14

graph. the on line double theby

shown as ofenergy mechanical total a has object the Suppose

xU

J5

move? object the can of values whatBetween d. x

totalE

. wherepositions reach cannot

object the positive always is the Since

JxU

Ek5

mxm

between moves object the Therefore,

105.2

xUEE Tk

2

2

1mvEk and

Consider an ideal spring. Elastic Potential Energy Hooke’s Law.

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Transcript of Consider an ideal spring. Elastic Potential Energy Hooke’s Law.