Transcript Potential Energy - ShareStudies.com

Chapter 8
Potential Energy & Conservation
of Energy
Potential Energy

Potential energy is energy related to the
configuration of a system in which the
components of the system interact by forces


The forces are internal to the system
Can be associated with only specific types of
forces acting between members of a system
Gravitational Potential Energy


The system is the Earth and
the book
Do work on the book by lifting
it slowly through a vertical
displacement
The work done on the system
must appear as an increase in the
energy of the system

Gravitational Potential Energy,
cont


There is no change in kinetic energy since the book
starts and ends at rest
Gravitational potential energy is the energy
associated with an object at a given location above
the surface of the Earth
Gravitational Potential Energy,
final




The quantity mgy is identified as the gravitational
potential energy, Ug
 Ug = mgy
Units are joules (J)
Is a scalar
Work may change the gravitational potential energy
of the system
 Wnet = DUg
Gravitational Potential Energy,
Problem Solving


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
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 kx 2
The elastic potential energy can
be thought of as the energy stored
in the deformed spring.
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
 x 2 will always be positive
Example:
An object moves in the xy-plane is subjected to a
force that does 14 J of work on it as it moves from
the origin to the position (2i - 3j) meters. Consider
the three possible forces:
F1 = 4i - 2j ;
F2 = i - 4j ;
F3 = 2i -3j
Which of the following statement is true?
a) F1 or F2 could have done the work.
b) Only F1 could have done the work.
c) F1 or F3 could have done the work.
d) Only F2 could have done the work.
e) Only F3 could have done the work.
Example:
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
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
Conservative Forces, cont

Examples of conservative forces:



Gravity
Spring force
We can associate a potential energy for a
system with any conservative force acting
between members of the system


This can be done only for conservative forces
In general: WC = - DU
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, cont


The work done against
friction is greater along
the brown path than
along the blue path
Because the work done
depends on the path,
friction is a
nonconservative force
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 done by such a force, F, is

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

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
 Can be extended to three dimensions
Conservative Forces and
Potential Energy – Check

Look at the case of a deformed spring


This is Hooke’s Law and confirms the equation for
U
U is an important function because a
conservative force can be derived from it
Example:
As a particle moves along the x
axis it is acted by a
conservative force. The
potential energy is shown as a
function of the coordinate x of
the particle. Rank the labeled
regions according to the
magnitude of the force, least to greatest.
1) AB, CD, BC
3) CD, BC, AB
2) AB, BC, CD
4) BC, AB, CD
Potential Energy in Molecules

There is potential energy associated with the
force between two neutral atoms in a
molecule which can be modeled by the
Lennard-Jones function
Potential Energy Curve of a
Molecule


Find the minimum of the function (take the derivative and set
it equal to 0) to find the separation for stable equilibrium
The graph of the Lennard-Jones function shows the most
likely separation between the atoms in the molecule (at
minimum energy)
Energy Review

Kinetic Energy


Potential Energy



Associated with movement of members of a
system
Determined by the configuration of the system
Gravitational and Elastic
Internal Energy

Related to the temperature of the system
Types of Systems

Nonisolated systems



Energy can cross the system boundary in a
variety of ways
Total energy of the system changes
Isolated systems


Energy does not cross the boundary of the
system
Total energy of the system is constant
Ways to Transfer Energy Into
or Out of A System



Work – transfers by applying a
force and causing a displacement
of the point of application of the
force
Mechanical Waves – allow a
disturbance to propagate through
a medium
Heat – is driven by a temperature
difference between two regions in
space
More Ways to Transfer Energy
Into or Out of A System

Matter Transfer – matter
physically crosses the boundary
of the system, carrying energy
with it

Electrical Transmission –
transfer is by electric current

Electromagnetic Radiation –
energy is transferred by
electromagnetic waves
Conservation of Energy

Energy is conserved


This means that energy cannot be created nor
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
Conservation of Energy, cont.

Mathematically, DEsystem = ST


Esystem is the total energy of the system
T is the energy transferred across the system
boundary
Established symbols: Twork = W and Theat = Q
Others just use subscripts
Conservation of Energy, cont.
The Work-Kinetic Energy theorem (DKE = W )
is a special case of Conservation of Energy

The full expansion of the above equation gives
D K + D U + DEint = W + Q + TMW + TMT + TET +
TER
Isolated System
In any isolated system of objects that
interact only through conservative forces,
the total mechanical energy of the system
remains constant.
ΔEm ec h  0  Ei,m ec h  E f ,m ec h
K i  Ui  K f  Uf
Remember: Total mechanical energy is the sum of the
kinetic and potential energies in the system
Isolated System
If nonconservative (nc) forces are acting, some
energy is transformed into internal energy
Conservation of Energy becomes:
DEsystem = 0
Wnc = DEmech
Wnc  (K f  Uf )  (K i  Ui )
Esystem is all kinetic, potential, and internal
energies
This is the most general statement of the
isolated system model
Problem Solving Strategy –
Conservation of Mechanical Energy for
an Isolated System

Categorize

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Define the system
It may consist of more than one object and may or
may not include springs or other sources of
storing potential energy
Determine if any energy transfers occur across
the boundary of your system


If there are transfers, use DEsystem = ST
If there are no transfers, use DEsystem = 0
Problem Solving Strategy, cont
Determine is there are any nonconservative
forces acting

If not, use the principle of conservation of mechanical
energy
Problem-Solving Strategy, 2
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Analyze

Choose configurations to represent initial and final
configuration of the system

For each object that changes height, identify the
zero configuration for gravitational potential energy

For each object on a spring, the zero configuration
for elastic potential energy is when the object is in
equilibrium
Problem-Solving Strategy, 3

Analyze, cont


Write expressions for total initial energy and total
final energy
Use and solve the appropriate conservation of
Energy Equation
Example – Free Fall

Determine the speed of
the ball at y above the
ground
Example – Free Fall, cont
Categorize
System is isolated Only force is
gravitational which is
conservative
Analyze


Apply Conservation of Energy
Kf + Ugf = Ki + Ugi


Ki = 0, the ball is dropped
Solving for vf
v f  v i2  2g  h  y 
Example:


A pendulum consists of a
sphere of mass m attached to
a light cord of length L. The
sphere is released from rest at
point A when the cord makes
an angle qA with the vertical,
and the pivot at point P is
frictionless.
Find the speed of the sphere
when it is at the lowest point B.
Example:
Two objects interact with each other and with no other
objects. Initially object A has a speed of 5 m/s and
object B has a speed of 10 m/s. In the course of their
motion they return to their initial positions. Then A has a
speed of 4 m/s and B has a speed of 7 m/s. We can
conclude:
A) the potential energy changed from the beginning to the end of the trip
B) mechanical energy was increased by conservative forces
C) mechanical energy was decreased by conservative forces
D) mechanical energy was decreased by nonconservative forces
E) mechanical energy was increased by nonconservative forces
Solution:

Assuming that there are no
friction. The mechanical energy
of the ball is conserved:
E mech, A  E mech, B
KA  UA  KB  UB
0
1
2
 mgh  mv B  0
2
But, h = L – L cos(qA)
 v B  2gL(1 cosq A )
h
U=0
Solution cont’d:
What is the tension TB in
the string at point B.
Newton’s Law:
2
vB
T - mg  m
L
 T  mg  2mg(1 - cosθ A )
 T  mg(3 - 2cos θ A )
T
mg
Example:
A particle is released from
rest at the point x = a and
moves along the x axis
subject to the potential
energy function U(x) shown.
The particle:
A) moves to infinity at varying speed
B) moves to x = b where it remains at rest
C) moves to a point to the left of x = e, stops and remains at rest
D) moves to a point to x = e, then moves to the left
E) moves to x = e and then to x = d, where it remains at rest
Example:

The launching mechanism of a toy
gun consists of a spring of unknown
spring constant. When the spring is
compressed 0.120 m, the gun, when
fired vertically, is able to launch a
35.0 g projectile to a maximum
height of 20.0 m above the position
of the projectile before firing.
Neglecting all resistive forces
determine the spring constant k?
Solution:

The mechanical energy of the
Projectile-Spring-Earth is
conserved:
(KE  U g  U s ) C  (KE  U g  U s ) A
0  mgh 0  0  0
1 2
 kx
2
2mgh
k
x2
2  0.035  9.8  20.0

 953 N/m
2
(0.120)
Solution Cont’d:

Find the speed of the projectile
as it moves through the
equilibrium position of the spring?
(KE  U g  U s ) B  (KE  U g  U s ) A
1
1 2
2
mv B  mgx  0  0  0  kx
2
2
kx 2
vB 
 2gx
m
953 (0.120)2
vB 
 2  9.8 0.120  19.7 m/s
0.035
Example:
Two block are connected by a
light string that passes over a
frictionless pulley, as shown.
The block of mass m1 lies on a
horizontal table and is connected
to a spring of constant k.
The system is released from rest when
the spring is unstretched.
If the hanging block of mass m2 falls a distance h before
coming to rest, calculate the coefficient of kinetic friction
between the table and the block.
Solution:
System = m1, m2, earth and spring
The Change in Mechanical energy
is equal to the work done by friction:
Wfriction = Ef – Ei = DEmech
Wf = -f h = -mk m1g h
Ef = ½ kh2 and Ei = m2gh
 -mk m1g h = ½ kh2 - m2gh
1
m2g - kh
2
μk 
m1g
N.B. in this example if we include
the table in our system:
DEsystem = 0; i.e. DEmech +DEint = 0
Example – Spring Mass
Collision


Without friction, the energy
continues to be transformed
between kinetic and elastic
potential energies and the
total energy remains the
same
If friction is present, the
energy decreases

DEmech = -ƒkd