Download Chapter 7 - UCF Physics

Subject to much change
Chapter 7
Energy
and
Energy Transfer
February 22, 2006
HAPPY BIRTHDAY
GEORGE!
Kalendar



Today we start the new TOPIC OF ENERGY
No Quiz on Friday, but there MAY be one on
Monday
The BAD NEWS:

EXAM #2 will be on March 3 (Friday)
ENERGY





We use energy to walk, run or even sleep
We use energy when we lift a weight
We use energy when we drive a car
We even use energy to THINK!
BUT …..
Introduction to Energy



The concept of energy is one of the
most important topics in science
Every physical process that occurs in
the Universe involves energy and
energy transfers or transformations
Energy is not easily defined
Systems

A system is a small portion of the Universe




We identify a number of particles or objects and
draw a sphere around them
There are no forces acting on anything inside the
sphere from outside the sphere
We will ignore the details outside of the sphere.
A critical skill is to identify the system
Valid System

A valid system may




be a single object or particle
be a collection of objects or particles
be a region of space
vary in size and shape
Environment

There is a system boundary around the
system



The boundary is an imaginary surface
It does not necessarily correspond to a physical
boundary
The boundary divides the system from the
environment

The environment is “the rest of the Universe”
Work

The work, W, done on a system by an agent
exerting a constant force on the system is the
product of the magnitude, F, of the force, the
magnitude Dr of the displacement of the point
of application of the force, and cos q, where q
is the angle between the force and the
displacement vectors
Working, Working, Working
WORK = Component of the applied force x
the displacement=
Fcos(q) x Dr
Work, cont.

W = F Dr cos q



 F  Dr
(Later for the dot!)
The displacement is that of the point of application of
the force
A force does no work on the object if the force does not
move through a displacement
The work done by a force on a moving object is zero
when the force applied is perpendicular to the
displacement of its point of application
Work Example

The normal force, n,
and the gravitational
force, m g, do no work
on the object



cos q = cos 90° = 0
The force F does do
work on the object
Same amount as in the
previous overhead
More About Work

The system and the environment must be
determined when dealing with work

The environment does work on the system


Work by the environment on the system
The sign of the work depends on the direction of F
relative to Dr


Work is positive when projection of F onto Dr is in the same
direction as the displacement
Work is negative when the projection is in the opposite
direction
Units of Work


Work is a scalar quantity
The unit of work is a joule (J)


1 joule = 1 newton . 1 meter
J=N·m
A block of mass 2.50 kg is pushed 2.20 m along a
frictionless horizontal table by a constant 16.0-N
force directed 25.0 below the horizontal.
Determine the work done on the block by (a) the
applied force, (b) the normal force exerted by the
table, and (c) the gravitational force. (d)
Determine the total work done on the block.
A raindrop of mass 3.35  10–5 kg falls vertically at
constant speed under the influence of gravity and air
resistance. Model the drop as a particle. As it falls
100 m, what is the work done on the raindrop (a) by
the gravitational force and (b) by air resistance?
Work Is An Energy Transfer



This is important for a system approach to
solving a problem
If the work is done on a system and it is
positive, energy is transferred to the system
If the work done on the system is negative,
energy is transferred from the system
Work Is An Energy Transfer, cont

If a system interacts with its environment, this
interaction can be described as a transfer of
energy across the system boundary

This will result in a change in the amount of
energy stored in the system
Shejule




Continue to work on energy.
Exam on March 3rd.
Material … as far as we get by March
1st.
Mucho WebAssign Stuff
LAST TIME




We defined the “Dot Product”
We defined WORK
We discussed the “system” and the
“environment”
System + Environment = Entire
Universe (Dumb concept!)
WORK
W  F  Dr  FDr cos(q )
F is the NET force acting on the block.
Scalar or DOT Product of Two Vectors

The scalar product of
two vectors is written
as A . B


It is also called the dot
product
A . B = A B cos q

q is the angle between A
and B
Scalar Product Properties

The scalar product is commutative



A .B = B .A
The scalar product obeys the distributive law
of multiplication
A . (B + C) = A . B + A . C
Dot Products of Unit Vectors

î  î  ĵ  ĵ  k̂  k̂  1
î  ĵ  î  k̂  ĵ  k̂  0

Using component form with A and B:
A  A x î  A y ĵ  A zk̂
B  B x î  B y ĵ  B zk̂
A  B  A xB x  A yB y  A zB z
A force



F  6 ˆi – 2 ˆj N
acts on a particle that undergoes a displacement .

Dr  3 ˆi  ˆj m
Find (a) the work done by the force on the particle and (b) the angle between
F and r.
Work Done by a Varying Force
xf
W   Fx Dx
xi
xf
W   Fx ( x)dx
xi
11.
The force acting on a particle varies as in Figure P7.11. Find the
work done by the force on the particle as it moves (a) from
x = 0 to x = 8.00 m, (b) from x = 8.00 m to
x = 10.0 m, and (c) from x = 0 to x = 10.0 m.
Work Done By Multiple Forces

AGAIN … If more than one force acts on a
system and the system can be modeled as a
particle, the total work done on the system is
the work done by the net force
W  W
net

xf
xi
  F dx
x
Hooke’s Law

The force exerted by the spring is
Fs = - kx



x is the position of the block with respect to the equilibrium
position (x = 0)
k is called the spring constant or force constant and measures
the stiffness of the spring
This is called Hooke’s Law
Hooke’s Law, cont.



When x is positive
(spring is stretched), F
is negative
When x is 0 (at the
equilibrium position), F
is 0
When x is negative
(spring is compressed),
F is positive
Work Done by a Spring


Identify the block as the
system
Calculate the work as the
block moves from xi = xmax to xf = 0
xf
0
xi
 xmax
Ws   Fx dx  

 kx  dx 
1 2
kxmax
2
The total work done as the
block moves from
–xmax to xmax is zero
ENERGY IS STORED IN THE SPRING AND THEN RECOVERED AND THEN
STORED AND THEN RECOVERED AND THEN STORED AND THEN
RECOVERED AND THEN STORED AND THEN RECOVERED AND THEN ….
Spring with an Applied Force




Suppose an external agent,
Fapp, stretches the spring
The applied force is equal
and opposite to the spring
force
Fapp = -Fs = -(-kx) = kx
Work done by Fapp is equal
to ½ kx2max
19.
If it takes 4.00 J of work to stretch a
Hooke's-law spring 10.0 cm from its
unstressed length, determine the extra work
required to stretch it an additional 10.0 cm.
21.
A light spring with spring constant
1 200 N/m is hung from an elevated support. From its lower end a second
light spring is hung, which has spring constant
1 800 N/m. An object of mass 1.50 kg is hung at rest from the lower end
of the second spring. (a) Find the total extension distance of the pair of
springs. (b) Find the effective spring constant of the pair of springs as a
system. We describe these springs as in series.
Consider the following:
xf
W   Fdx
xi
xf
F
xf
dv
W   madx   m dx
dt
xi
xi
xf
vf
dx
W   mdv   mvdv
dt vi
xi
1 2 1 2
W  mv f  mvi
2
2
Kinetic Energy

Kinetic Energy is the energy
of a particle due to its
motion

K = ½ mv2




K is the kinetic energy
m is the mass of the
particle
v is the speed of the
particle
A change in kinetic energy
is one possible result of
doing work to transfer
energy into a system
1 2 1 2
W  mv f  mvi
2
2
With no friction or other strange forces,
the work done by a force on a particle
(or system of particles) = the change in
the particles kinetic energy.
Work-Kinetic Energy Theorem



The Work-Kinetic Energy Principle states SW = Kf –
Ki = DK
In the case in which work is done on a system and
the only change in the system is in its speed, the
work done by the net force equals the change in
kinetic energy of the system.
We can also define the kinetic energy

K = ½ mv2
Nonisolated System

A nonisolated system is one that interacts
with or is influenced by its environment


An isolated system would not interact with its
environment
The Work-Kinetic Energy Theorem can be
applied to nonisolated systems
BREAK POINT
February 27, 2006
Stuff Happens

Today and Wednesday
– More on potential energy
– some material from the next chapter

Friday
– EXAMINATION #2
– All material since last exam.
– You should be studying by now.
Internal Energy



The energy associated with
an object’s temperature is
called its internal energy,
Eint
In this example, the surface
is the system
The friction does work and
increases the internal
energy of the surface
FRICTION
Potential Energy


Potential energy is energy related to the
configuration of a system in which the
components of the system interact by
forces
Examples include:



elastic potential energy – stored in a spring
gravitational potential energy
electrical potential energy
Conservation of Energy

Energy is conserved !!!!


This means that energy cannot be created or
destroyed
If the total amount of energy in a system changes,
it can only be due to the fact that energy has
crossed the boundary of the system by some
method of energy transfer
A 2 100-kg pile driver is used to drive a steel I-beam into the
ground. The pile driver falls 5.00 m before coming into contact with
the top of the beam, and it drives the beam 12.0 cm farther into the
ground before coming to rest. Using energy considerations,
calculate the average force the beam exerts on the pile driver while
the pile driver is brought to rest.
31.
A 40.0-kg box initially at rest is pushed 5.00 m along a
rough, horizontal floor with a constant applied horizontal force of
130 N. If the coefficient of friction between box and floor is 0.300,
find (a) the work done by the applied force, (b) the increase in
internal energy in the box-floor system due to friction, (c) the work
done by the normal force, (d) the work done by the gravitational
force, (e) the change in kinetic energy of the box, and (f) the final
speed of the box.
63.
The ball launcher in a pinball machine has a spring
that has a force constant of 1.20 N/cm (Fig. P7.63). The surface on
which the ball moves is inclined 10.0° with respect to the horizontal.
If the spring is initially compressed 5.00 cm, find the launching
speed of a 100-g ball when the plunger is released. Friction and the
mass of the plunger are negligible.
32.
A 2.00-kg block is attached to a spring of force constant
500 N/m as in Figure 7.10. The block is pulled 5.00 cm to the
right of equilibrium and released from rest. Find the speed of the
block as it passes through equilibrium if (a) the horizontal
surface is frictionless and (b) the coefficient of friction between
block and surface is 0.350.
35.
A sled of mass m is given a kick on a frozen pond.
The kick imparts to it an initial speed of 2.00 m/s. The coefficient
of kinetic friction between sled and ice is 0.100. Use energy
considerations to find the distance the sled moves before it stops.
Power


The time rate of energy transfer is called
power
The average power is given by
W
P
Dt
when the method of energy transfer is work
Instantaneous Power

The instantaneous power is the limiting
value of the average power as Dt approaches
zero
P

lim
Dt 0
W dW

Dt
dt
This can also be written as
dW
dr
P
 F   F v
dt
dt
Power Generalized


Power can be related to any type of energy transfer
In general, power can be expressed as
dE
P
dt

dE/dt is the rate rate at which energy is crossing the
boundary of the system for a given transfer
mechanism
Units of Power

The SI unit of power is called the watt


A unit of power in the US Customary system
is horsepower


1 watt = 1 joule / second = 1 kg . m2 / s2
1 hp = 746 W
Units of power can also be used to express
units of work or energy

1 kWh = (1000 W)(3600 s) = 3.6 x106 J
49.
A 4.00-kg particle moves along the x axis. Its position
varies with time according to x = t + 2.0t3, where x is in meters
and t is in seconds. Find (a) the kinetic energy at any time t,
(b) the acceleration of the particle and the force acting on it at
time t, (c) the power being delivered to the particle at time t,
and (d) the work done on the particle in the interval t = 0 to t =
2.00 s.
BREAK
EXAM #2 This Friday!
Let’s Review with some
Problems
24.
A 5.00-kg object placed on a frictionless, horizontal
table is connected to a cable that passes over a pulley and
then is fastened to a hanging 9.00-kg object, as in Figure
P5.24. Draw free-body diagrams of both objects. Find the
acceleration of the two objects and the tension in the string.
In Figure P5.29, the
man and the platform
together weigh 950 N.
The pulley can be
modeled as frictionless.
Determine how hard the
man has to pull on the
rope to lift himself
steadily upward above
the ground. (Or is it
impossible? If so,
explain why.)
34.
An object of mass m1 on a frictionless horizontal table is
connected to an object of mass m2 through a very light pulley P1 and
a light fixed pulley P2 as shown in Figure P5.34. (a) If a1 and a2 are
the accelerations of m1 and m2, respectively, what is the relation
between these accelerations? Express (b) the tensions in the strings
and (c) the accelerations a1 and a2 in terms of the masses m1, and
m2, and g.
44.
Three objects are connected on the table as shown in Figure P5.44.
The table is rough and has a coefficient of kinetic friction of 0.350. The
objects have masses 4.00 kg, 1.00 kg and 2.00 kg, as shown, and the
pulleys are frictionless. Draw free-body diagrams of each of the objects. (a)
Determine the acceleration of each object and their directions. (b) Determine
the tensions in the two cords.
68.
Two blocks of mass 3.50 kg and 8.00 kg are connected by
a massless string that passes over a frictionless pulley (Fig. P5.68).
The inclines are frictionless. Find (a) the magnitude of the
acceleration of each block and (b) the tension in the string.
19.
A roller coaster car (Fig. P6.19) has a mass of 500 kg when fully
loaded with passengers. (a) If the vehicle has a speed of 20.0 m/s at
point A, what is the force exerted by the track on the car at this point? (b)
What is the maximum speed the vehicle can have at B and still remain on
the track?
25. A person stands on a scale in an
elevator. As the elevator starts, the scale
has a constant reading of 591 N. As the
elevator later is stopping, the scale
reading is 391 N. Assume the magnitude
of the acceleration is the same during
starting and stopping, and determine (a)
the weight of the person, (b) the person's
mass, and (c) the acceleration of the
elevator.
7.
A crate of eggs is located in the middle of
the flat bed of a pickup truck as the truck
negotiates an unbanked curve in the road. The
curve may be regarded as an arc of a circle of
radius 35.0 m. If the coefficient of static friction
between crate and truck is 0.600, how fast can
the truck be moving without the crate sliding?
8.
Find the scalar product of the vectors in Figure P7.8.
15.
When a 4.00-kg object is hung vertically on a
certain light spring that obeys Hooke's law, the spring
stretches 2.50 cm. If the 4.00-kg object is removed, (a)
how far will the spring stretch if a 1.50-kg block is hung
on it, and (b) how much work must an external agent
do to stretch the same spring 4.00 cm from its
unstretched position?
20.
A small particle of mass m is pulled to the top of a frictionless halfcylinder (of radius R) by a cord that passes over the top of the cylinder, as
illustrated in Figure P7.20. (a) If the particle moves at a constant speed, show
that F = mgcos. (Note: If the particle moves at constant speed, the component
of its acceleration tangent to the cylinder must be zero at all times.) (b) By
directly integrating W = Fdr, find the work done in moving the particle at
constant speed from the bottom to the top of the half-cylinder.
49.
A 4.00-kg particle moves along the x axis. Its
position varies with time according to x = t + 2.0t3, where x
is in meters and t is in seconds. Find (a) the kinetic energy
at any time t, (b) the acceleration of the particle and the
force acting on it at time t, (c) the power being delivered to
the particle at time t, and (d) the work done on the particle
in the interval t = 0 to t = 2.00 s.
Potential Energy
~March 7, 2006
Potential (Stored) Energy

Potential energy is the energy associated with the
configuration of a system of objects that exert forces
on each other

This can be used only with conservative forces


Conservative forces are NOT Republicans
When conservative forces act within an isolated system,
the kinetic energy gained (or lost) by the system as its
members change their relative positions is balanced by an
equal loss (or gain) in potential energy.

This is Conservation of Mechanical Energy
Types of Potential Energy



There are many forms of potential energy, including:
 Gravitational
 Electromagnetic
 Chemical
 Nuclear
One form of energy in a system can be converted into another
 Nuclear heat easy
 Heat Nuclear probably impossible!
Conversion from one type to another type of energy is not always
reversible.
Systems with Multiple Particles



We can extend our definition of a system to
include multiple objects
The force can be internal to the system
The kinetic energy of the system is the
algebraic sum of the kinetic energies of the
individual objects

Sometimes, the kinetic energy of one of the
objects may be negligible
System Example








This system consists of Earth and a
book
Do work on the system by lifting the
book through Dy
The work done by you is mg(yb – ya)
At the top it is at rest.
The amount of work that you did is
called the potential energy of the
system with respect to the ground.
The PE’s initial value is mgya
The FINAL value is mgyb
The difference is the work done.
Let’s drop the book from yb and see what
it is doing at ya.
vv2  v02  2 g ( yb  ya )
v 0
2
0
Multiply by m and divide by 2:
1 2
mv f  mg ( yb  ya )
2
The decrease in Potential Energy equals the increase in Kinetic Energy
Potential Energy



The energy storage mechanism is called
potential energy
A potential energy can only be associated
with specific types of forces (conservative)
Potential energy is always associated with a
system of two or more interacting objects
Gravitational Potential Energy



Gravitational Potential Energy is associated
with an object at a given distance above
Earth’s surface
Assume the object is in equilibrium and
moving at constant velocity
The work done on the object is done by Fapp
and the upward displacement is
Dr  Dyˆj
Gravitational Potential Energy, cont
W   Fapp   Dr

W  (mgˆj)   yb  ya  ˆj
W  mgyb  mgya

The quantity mgy is identified as the
gravitational potential energy, Ug


Ug = mgy
Units are joules (J)
Energy Problems





Draw a diagram of the situation.
ESTABLISH AN ORIGIN.
THE POTENTIAL ENERGY OF A PARTICAL
OF MASS M IS ALWAYS MEASURED WITH
RESPECT TO THIS ORIGIN.
The potential energy of a particle is defined
as being ZERO when it is at the origin.
At some height above the origin, the value of
the PE is mgh.
Gravitational Potential Energy, final


The gravitational potential energy depends only
on the vertical height of the object above Earth’s
surface
In solving problems, you must choose a
reference configuration for which the gravitational
potential energy is set equal to some reference
value, normally zero

The choice is arbitrary because you normally need
the difference in potential energy, which is
independent of the choice of reference
configuration
Conservation of Mechanical Energy

The mechanical energy of a system is the algebraic
sum of the kinetic and potential energies in the
system


Emech = K + Ug
The statement of Conservation of Mechanical
Energy for an isolated system is Kf + Uf = Ki+ Ui

An isolated system is one for which there are no energy
transfers across the boundary
A bead slides without friction around a loop-the-loop (Fig. P8.5). The
bead is released from a height h = 3.50R. (a) What is its speed at
point A? (b) How large is the normal force on it if its mass is 5.00 g?
Let’s look at the more general case.
y=h
m
W=0
d
D
F
y=0
(origin)
W=mgD
We do work to
move mass to
y=h. W=mgh
m
m
ANY PATH


Can be broken up into a series of very small
vertical moves and horizontal moves.
The horizontal moves require no work.


The force is at right angles to the motion. Dot product is
zero.
The vertical moves are
W   FD  F  D  Fh  mgh
Conservation of Mechanical Energy,
example





Look at the work done by
the book as it falls from
some height to a lower
height
Won book = DKbook
Also, W = mgyb – mgya
So, DK = -Dug
Or: DK +Dug=0
Elastic Potential Energy



Elastic Potential Energy is associated with a
spring
The force the spring exerts (on a block, for example)
is Fs = - kx
The work done by an external applied force on a
spring-block system is


W = ½ kxf2 – ½ kxi2
The work is equal to the difference between the initial and
final values of an expression related to the configuration of
the system
Elastic Potential Energy, cont



This expression is the
elastic potential energy: Us
= ½ kx2
The elastic potential energy
can be thought of as the
energy stored in the
deformed spring
The stored potential energy
can be converted into
kinetic energy
Elastic Potential Energy, final

The elastic potential energy stored in a spring is
zero whenever the spring is not deformed (U = 0
when x = 0)



The energy is stored in the spring only when the spring is
stretched or compressed
The elastic potential energy is a maximum when the
spring has reached its maximum extension or
compression
The elastic potential energy is always positive

x2 will always be positive
The Bindell Conservation of Energy
Equation
The sum of the Kinetic Energy and
the potential energy of the system
before a sequence of events
Is Equal to
The sum of the Kinetic Energy,
the potential energy of the system
and the Energy Lost to friction or
witchcraft.
An object of mass m starts from rest and slides a distance d down a
frictionless incline of angle . While sliding, it contacts an unstressed
spring of negligible mass as shown in Figure P8.10. The object
slides an additional distance x as it is brought momentarily to rest by
compression of the spring (of force constant k). Find the initial
separation d between object and spring.
CAN WE ADD
FRICTION???
Problem Solving Strategy – Conservation of
Mechanical Energy

Define the isolated system and the initial
and final configuration of the system



The system may include two or more
interacting particles
The system may also include springs or other
structures in which elastic potential energy can
be stored
Also include all components of the system that
exert forces on each other
Problem-Solving Strategy, 2

Identify the configuration for zero potential
energy


Include both gravitational and elastic potential
energies
If more than one force is acting within the system,
write an expression for the potential energy
associated with each force
Problem-Solving Strategy, 3


If friction or air resistance is present,
mechanical energy of the system is not
conserved
Use energy with non-conservative forces
instead
Problem-Solving Strategy, 4

If the mechanical energy of the system is
conserved, write the total energy as



Ei = Ki + Ui for the initial configuration
Ef = Kf + Uf for the final configuration
Since mechanical energy is conserved, Ei =
Ef and you can solve for the unknown
quantity
Two objects are connected by a light string passing over a light
frictionless pulley as in Figure P8.13. The object of mass m1 is
released from rest at height h. Using the principle of conservation of
energy, (a) determine the speed of m2 just as m1 hits the ground. (b)
Find the maximum height to which m2 rises.
A particle of mass m = 5.00 kg is released from point A and slides on the
frictionless track shown in Figure P8.24. Determine (a) the particle's speed
at points B and C and (b) the net work done by the gravitational force in
moving the particle from A to C.
A block slides down a curved frictionless track and then up an inclined
plane as in Figure P8.48. The coefficient of kinetic friction between
block and incline is k. Use energy methods to show that the maximum
height reached by the block is
y max 
h
1   k cot q
A 10.0-kg block is released from point A in Figure P8.57. The track is
frictionless except for the portion between points B and C , which has a
length of
6.00 m. The block travels down the track, hits a spring of force constant 2
250 N/m, and compresses the spring 0.300 m from its equilibrium position
before coming to rest momentarily. Determine the coefficient of kinetic
friction between the block and the rough surface between B and C.
Conservation of Energy, (Pendulum)




As the pendulum swings, there
is a continuous change
between potential and kinetic
energies
At A, the energy is potential
At B, all of the potential energy
at A is transformed into kinetic
energy
 Let zero potential energy be
at B
At C, the kinetic energy has
been transformed back into
potential energy
Conservative Forces


The work done by a conservative force on a
particle moving between any two points is
independent of the path taken by the particle
The work done by a conservative force on a
particle moving through any closed path is
zero

A closed path is one in which the beginning and
ending points are the same
Nonconservative Forces


A nonconservative force does not satisfy the
conditions of conservative forces
Nonconservative forces acting in a system
cause a change in the mechanical energy of
the system
Nonconservative Forces


The work done against
friction is greater along
the red path than along
the blue path
Because the work done
depends on the path,
friction is a
nonconservative force
Nonconservative Forces (Connected
Blocks)



The system consists of
the two blocks, the spring,
and Earth
Gravitational and potential
energies are involved
The kinetic energy is zero
if our initial and final
configurations are at rest
Connected Blocks, cont



Block 2 undergoes a change in gravitational
potential energy
The spring undergoes a change in elastic
potential energy
The coefficient of kinetic energy can be
measured
Conservative Forces and Potential
Energy


Define a potential energy function, U, such
that the work done by a conservative force
equals the decrease in the potential energy of
the system
The work donex by such a force, F, is
WC   Fx dx  DU
f
xi

DU is negative when F and x are in the same
direction
Conservative Forces and Potential
Energy

The conservative force is related to the potential
energy function through
dU
Fx  
dx

The x component of a conservative force acting on
an object within a system equals the negative of the
potential energy of the system with respect to x
Conservative Forces and Potential
Energy – Check

Look at the case of a deformed spring
dU s
d 1 2
Fs  
   kx   kx
dx
dx  2


This is Hooke’s Law
Energy Diagrams and Equilibrium



Motion in a system can be observed in terms of a graph of its
position and energy
In a spring-mass system example, the block oscillates
between the turning points, x = ±xmax
The block will always accelerate back toward x = 0
Energy Diagrams and Stable
Equilibrium



The x = 0 position is one of
stable equilibrium
Configurations of stable
equilibrium correspond to
those for which U(x) is a
minimum
x=xmax and x=-xmax are
called the turning points
Energy Diagrams and Unstable
Equilibrium




Fx = 0 at x = 0, so the
particle is in equilibrium
For any other value of x, the
particle moves away from
the equilibrium position
This is an example of
unstable equilibrium
Configurations of unstable
equilibrium correspond to
those for which U(x) is a
maximum