Download potential energy.

Chapter 8
Potential Energy and Conservation of Energy
Potential Energy
Potential energy is energy determined by 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.
 If the arrangement of the system changes, then the potential energy of the
system changes
Prof Dr Ahmet ATAÇ
Section 7.6
Gravitational Potential Energy
As an object falls toward the Earth, the
Earth exerts a gravitational force mg on
the object, with the direction of the force
being the same as the direction of the
object’s motion. For example;
The system is the Earth and the book.
Do work on the book by lifting it slowly
through a vertical displacement.
r   y  y  ˆj
f
i
The work done on the system must appear
as an increase in the energy of the system.
The energy storage mechanism is called
potential energy.
Prof Dr Ahmet ATAÇ
Section 7.6
Gravitational Potential Energy, cont
Assume the book in fig. is allowed to fall.
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.
 
Wext  Fapp   r
Wext  (mgˆj)   y f  y i  ˆj
Wext  mgy f  mgy i
Prof Dr Ahmet ATAÇ
Section 7.6
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.
 Wext = ug
Potential energy is always associated with a system of two or more interacting
objects.
Prof Dr Ahmet ATAÇ
Section 7.6
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.
Often having the object on the surface of the Earth is a convenient zero
gravitational potential energy configuration.
The problem may suggest a convenient configuration to use.
Prof Dr Ahmet ATAÇ
Section 7.6
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.
Prof Dr Ahmet ATAÇ
Section 7.6
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.
Observe the effects of different amounts of
compression of the spring.
Prof Dr Ahmet ATAÇ
Section 7.6
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.
Prof Dr Ahmet ATAÇ
Section 7.6
Prof Dr Ahmet ATAÇ
Conservatıve and Nonconservatıve Forces
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.
Examples of conservative forces:
 Gravity
 Spring force
Prof Dr Ahmet ATAÇ
Conservative Forces, cont
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: Wint = - U
 Wint is used as a reminder that the work is done by one member of the system on
another member and is internal to the system.
 Positive work done by an outside agent on a system causes an increase in
the potential energy of the system.
 Work done on a component of a system by a conservative force internal to
an isolated system causes a decrease in the potential energy of the system.
Prof Dr Ahmet ATAÇ
Section 7.7
Non-conservative Forces
A non-conservative force does not satisfy the conditions of conservative forces.
Non-conservative forces acting in a system cause a change in the mechanical
energy of the system.
Emech = K + U
 K includes the kinetic energy of all moving members of the system.
 U includes all types of potential energy in the system.
Prof Dr Ahmet ATAÇ
Section 7.7
Non-conservative 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 non-conservative
force.
Prof Dr Ahmet ATAÇ
Section 7.7
Example 1
A 40.0-N child is in a swing that is attached to ropes 2.00 m long. Find the
gravitational potential energy of the child – Earth system relative to the child’s
lowest position when (a) the ropes are horizontal, (b) the ropes make a 30.0°
angle with the vertical, and (c) the child is at the bottom of the circular arc
Prof Dr Ahmet ATAÇ
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
xf
Wint   Fx dx  U
xi
 U is negative when F and x are in the same direction.
Prof Dr Ahmet ATAÇ
Section 7.8
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.
 Can be extended to three dimensions
Prof Dr Ahmet ATAÇ
Section 7.8
Conservative Forces and Potential Energy – Check
Look at the case of a deformed spring:
Us=1/2kx2
Fs  
dUs
d 1 2

kx   kx
dx
dx  2

 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.
Prof Dr Ahmet ATAÇ
Solution 1
Prof Dr Ahmet ATAÇ
Example 2
A 4.00-kg particle moves from the origin to position C, which has coordinates
x=5.00 m and y=5.00 m (Fig.). One force on it is the force of gravity acting in the
negative y direction. Calculate the work done by gravity as the particle moves
from O to C along (a) OAC, (b) OBC, and (c) OC. Your results should all be
identical. Why?
Prof Dr Ahmet ATAÇ
Solution 2
The results should all be the same, since gravitational forces are conservative.
Prof Dr Ahmet ATAÇ
Conservatıon of Mechanıcal Energy
An object in free fall, this principle tells us that any increase (or decrease) in
potential energy is accompanied by an equal de crease (or increase) in kinetic
energy. Note that the total mechanical energy of a system remains constant in
any isolated system of objects that interact only through conservative forces.
Total mechanical energy E of a system is defined as the sum of the kinetic and
potential energies, we can write; E K+ U
We can state the principle of conservation of energy Ei=Ef as and so we have
Ki + Ui = Uf + Kf
Prof Dr Ahmet ATAÇ
Example
At time ti, the kinetic energy of a particle is 30.0 J and the potential energy of
the system to which it belongs is 10.0 J. At some later time tf, the kinetic energy
of the particle is 18.0 J. (a) If only conservative forces act on the particle, what
are the potential energy and the total energy at time tf ? (b) If the potential energy
of the system at time tf is 5.00 J, are there any non-conservative forces acting on
the particle? Explain
Prof Dr Ahmet ATAÇ
Solution
Prof Dr Ahmet ATAÇ
Example
A single conservative force acts on a 5.00-kg particle. The equation
Fx=(2x+4) N, where x is in meters, describes this force. As the particle moves
along the x axis from x=1.00m to x=5.00m, calculate (a) the work done by this
force, (b) the change in the potential energy of the system, and (c) the kinetic
energy of the particle at x=5.00m if its speed at x=1.00m is 3.00 m/s.
Prof Dr Ahmet ATAÇ
Solution
Prof Dr Ahmet ATAÇ
Work done by Nonconservatıve Forces
if some of the forces acting on objects within the system are not conservative,
then the mechanical energy of the system does not remain constant. There are
two types of non-conservative forces:
• An applied force and
• The force of kinetic friction.
Work Done by an Applied Force
The force you apply does work Wapp on the book, while the gravitational force
does work Wg on the book. net work done on the book is related to the change in
its kinetic energy as described by the work – kinetic energy theorem given by
Equation; Wapp + Wg=K
Because the gravitational force is conservative
Wapp = K + U
We conclude that if an object is part of a system, then an applied force can
transfer energy into or out of the system.
Situations Involving Kinetic Friction
As the book moves through a distance d, the only force that does work is the
force of kinetic friction. This force causes a decrease in the kinetic energy of the
book;
Kfriction = fkd
If the book moves on an incline that is not frictionless, a change in the
gravitational potential energy of the book – Earth system also occurs, and –fkd is
the amount by which the mechanical energy of the system changes because of
the force of kinetic friction. In such cases,
E = K + U= –fkd
Ei + E = Ef
Example
A 70.0-kg diver steps off a 10.0-m tower and drops straight down into the
water. If he comes to rest 5.00 m beneath the surface of the water, determine the
average resistance force exerted by the water on the diver.
Example
The coefficient of friction between the 3.00-kg block and the surface in Figure
is 0.400. The system starts from rest. What is the speed of the 5.00-kg ball
when it has fallen 1.50 m?
Solution
Solutions
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.
Consider the potential energy function for a block – spring system, given by
Us=1/2kx2
Prof Dr Ahmet ATAÇ
Section 7.9
Energy Diagrams and Stable Equilibrium
The x = 0 position is one of stable
equilibrium.
 Any movement away from this
position results in a force directed
back toward x = 0.
Configurations of stable equilibrium
correspond to those for which U(x)
is a minimum.
x = xmax and x = -xmax are called the
turning points.
Prof Dr Ahmet ATAÇ
Section 7.9
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.
Prof Dr Ahmet ATAÇ
Section 7.9
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.
  12   6 
U ( x )  4       
 x  
 x 
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).
Prof Dr Ahmet ATAÇ
Section 7.9