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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
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 spring: Law sHooke' xkFs
dxkxU s 2
2
1kxEU es
JU
mx
The Potential Energy Function & Equilibrium
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
graph the of slope xUdx
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.
The Potential Energy Function & Equilibrium
JU
mx
graph the of slope xUdx
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.
The Potential Energy Function & Equilibrium
JU
mx
graph the of slope xUdx
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.
The Potential Energy Function & Equilibrium
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
graph the of slope xUdx
dUF
Nm
J
24
8
Nm
J
14
4
a. Sketch the corresponding Force vs. Position graph.
The Potential Energy Function & Equilibrium
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
b. Identify all points of equilibrium.
mx 5Stable Equilibrium
mxm 1411 Neutral Equilibrium
The Potential Energy Function & Equilibrium
Example An object has the potential energy function as given below.
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
The Potential Energy Function & Equilibrium
Example An object has the potential energy function as given below.
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