Transcript 6-5 Conservative and Nonconservative Forces Potential energy can

Chapter 6
Work and Energy
Objectives: The student will be
able to:
1.Define and calculate gravitational potential
energy.
2.State the work energy theorem and apply the
theorem to solve problems.
3.Distinguish between a conservative and a
non-conservative force and give examples of
each type of force.
6-4 Potential Energy
An object can have potential energy by virtue of
its surroundings.
Familiar examples of potential energy:
• A wound-up spring
• A stretched elastic band
• An object at some height above the ground
How is all energy divided?
All Energy
Potential
Energy
Gravitation
Potential
Energy
Elastic
Potential
Energy
Kinetic
Energy
Chemical
Potential
Energy
What is Potential Energy?
o Energy that is stored
and waiting to be
used later
What is Gravitational Potential
Energy?
o Potential energy
due to an object’s
position
Don’t look down,
Rover!
Good boy!
o P.E. = mass x
height x gravity
What is Elastic Potential Energy?
o Potential energy due compression or
expansion of an elastic object.
Notice the ball compressing
and expanding
What is Chemical Potential Energy?
o Potential energy
stored within the
chemical bonds of
an object
Which object has more potential energy?
B
A
ANSWER
A
This brick has more mass than the feather;
therefore more potential energy!
Changing an objects’ height can change its
potential energy.
If I want to drop an apple from the top of one of these
three things, where will be the most potential energy?
A
B
C
ANSWER
A
The higher the
object, the more
potential energy!
Roller Coasters
When does the train on
this roller coaster have
the MOST potential
energy?
AT THE VERY TOP!
The HIGHER the train is lifted
by the motor, the MORE
potential energy is produced.
At the top of the hill the train
has a huge amount of potential
energy, but it has very little
kinetic energy.
6-4 Potential Energy
In raising a mass m to a height
h, the work done by the
external force is
(6-5a)
We therefore define the
gravitational potential energy:
(6-6)
Potential Energy
• Potential energy is associated with the
position of the object
• Gravitational Potential Energy is the
energy associated with the relative
position of an object in space near the
Earth’s surface
• The gravitational potential energy
PE  mgy
– m is the mass of an object
– g is the acceleration of gravity
– y is the vertical position of the mass
relative the surface of the Earth
– SI unit: joule (J)
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Reference Levels
• A location where the gravitational potential
energy is zero must be chosen for each
problem
– The choice is arbitrary since the change in the
potential energy is the important quantity
– Choose a convenient location for the zero
reference height
• often the Earth’s surface
• may be some other point suggested by the problem
– Once the position is chosen, it must remain fixed
for the entire problem
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Work and Gravitational
Potential Energy
• PE = mgy
• Wg  F y cos   mg ( y f  yi ) cos180
 mg ( y f  yi )  PEi  PE f
• Units of Potential
Energy are the same
as those of Work and
Kinetic Energy
Wgravity  KE  PE  PEi  PE f
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6-4 Potential Energy
This potential energy can become kinetic energy
if the object is dropped.
Potential energy is a property of a system as a
whole, not just of the object (because it depends
on external forces).
If
, where do we measure y from?
It turns out not to matter, as long as we are
consistent about where we choose y = 0. Only
changes in potential energy can be measured.
6-4 Potential Energy
Potential energy can also be stored in a spring
when it is compressed; the figure below shows
potential energy yielding kinetic energy.
6-4 Potential Energy
The force required to
compress or stretch a
spring is:
(6-8)
where k is called the
spring constant, and
needs to be measured for
each spring.
Elastic Potential Energy:
Learning Goals
The student will describe the elastic potential energy of a
spring or similar object in qualitative and quantitative
terms and will investigate the transformation of
gravitational potential to elastic potential.
Elastic Potential Energy
Hooke’s Law
The stretch or compression of an elastic device
(e.g. a spring) is directly proportional to the
applied force:
Hooke’s Law
The stretch or compression of an elastic device
(e.g. a spring) is directly proportional to the
applied force:
Fx
stretch x 
or Fx  kx
k
x  0 is the equilibrium position
The spring constant
The constant k is called the spring constant or
force constant. It has units of N/m
and is the slope of the line in a force-extension
graph.
Example 1
A student stretches a spring 1.5 cm horizontally by
applying a force of magnitude 0.18 N.
Determine the force constant of the spring.
Example 1
A student stretches a spring 1.5 cm horizontally by
applying a force of magnitude 0.18 N.
Determine the force constant of the spring.
Givens :
x  0.015 m
F  0.18 N
Unknown :
k ?
Example 1
A student stretches a spring 1.5 cm horizontally by
applying a force of magnitude 0.18 N.
Determine the force constant of the spring.
Givens :
x  0.015 m
F  0.18 N
Unknown :
k ?
F
Select : F  kx  k 
x
0.18 N
Solve : k 
 12 Nm
0.015 m
Example 1
A student stretches a spring 1.5 cm horizontally by
applying a force of magnitude 0.18 N.
Determine the force constant of the spring.
Givens :
x  0.015 m
F  0.18 N
Unknown :
k ?
F
Select : F  kx  k 
x
0.18 N
Solve : k 
 12 Nm
0.015 m
Example 1
A student stretches a spring 1.5 cm horizontally by
applying a force of magnitude 0.18 N.
Determine the force constant of the spring.
Givens :
x  0.015 m
F  0.18 N
Unknown :
k ?
F
Select : F  kx  k 
x
0.18 N
Solve : k 
 12 Nm
0.015 m
Elastic Potential Energy
The force stretching or compressing a
spring is doing work on a spring,
increasing its elastic potential energy.
Note that this force is not constant but
increases linearly from 0 to kx. The
average force on the spring is ½kx.
Elastic Potential Energy
The force stretching or compressing a
spring is doing work on a spring,
increasing its elastic potential energy.
Note that this force is not constant but
increases linearly from 0 to kx. The
average force on the spring is ½kx.
W  F d  Fx x 
1
2
2

1
2
kx  x 
1
2
kx
2
This kx is the elastic potential energy , Ee .
Example 2
An apple of mass 0.10 kg is suspended from a vertical
spring with spring constant 9.6 N/m. How much elastic
potential energy is stored in the spring if the apple
stretches the spring 20.4 cm?
Givens :
k  9.6 Nm
x  0.204 m
Unknown :
Ee  ?
Example 2
An apple of mass 0.10 kg is suspended from a vertical
spring with spring constant 9.6 N/m. How much elastic
potential energy is stored in the spring if the apple
stretches the spring 20.4 cm?
Givens :
k  9.6 Nm
x  0.204 m
Unknown :
Ee  ?
Select : Ee  12 kx2
Solve : Ee 
1
2
9.6 0.204 m
Ee  0.20 J
N
m
2
Example 2
An apple of mass 0.10 kg is suspended from a vertical
spring with spring constant 9.6 N/m. How much elastic
potential energy is stored in the spring if the apple
stretches the spring 20.4 cm?
Givens :
k  9.6 Nm
x  0.204 m
Unknown :
Ee  ?
Select : Ee  12 kx2
Solve : Ee 
1
2
9.6 0.204 m
Ee  0.20 J
N
m
2
Example 2 Follow-Up
How much gravitational potential energy did the
apple lose?
Example 2 Follow-Up
How much gravitational potential energy did the
apple lose?
Givens :
m  0.10 kg
g  9.8 sm2
h  0.204 m
Unknown :
E g  ?
Example 2 Follow-Up
How much gravitational potential energy did the
apple lose?
Givens :
m  0.10 kg
g  9.8 sm2
h  0.204 m
Unknown :
E g  ?
Select : Eg  mgh


Solve : Eg  0.10 kg  9.8 sm2  0.204 m 
Eg  0.20 J
Example 2 Follow-Up
How much gravitational potential energy did the
apple lose?
Givens :
m  0.10 kg
g  9.8 sm2
h  0.204 m
Unknown :
E g  ?
Select : Eg  mgh


Solve : Eg  0.10 kg  9.8 sm2  0.204 m 
Eg  0.20 J
The ideal spring
An ideal spring is one that obeys Hooke’s Law –
within compression/stretching limits. Beyond
those limits the spring may deform.
The ideal spring
An ideal spring is one that obeys Hooke’s Law –
within compression/stretching limits. Beyond
those limits the spring may deform.
Be gentle with
my springs!
Potential Energy in a Spring
• Elastic Potential Energy:
1 2
PEs  kx
2
– SI unit: Joule (J)
– related to the work required to
compress a spring from its
equilibrium position to some final,
arbitrary, position x
• Work done by the spring
Ws  PEsi  PEsf
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6-4 Potential Energy
The force increases as the spring is stretched or
compressed further. We find that the potential
energy of the compressed or stretched spring,
measured from its equilibrium position, can be
written:
(6-9)
6-5 Conservative and Nonconservative
Forces
If friction is present, the work done depends not
only on the starting and ending points, but also
on the path taken. Friction is called a
nonconservative force.
Types of Forces
• Conservative forces
– Work and energy associated
with the force can be recovered
– Examples: Gravity, Spring Force,
EM forces
• Nonconservative forces
– The forces are generally
dissipative and work done
against it cannot easily be
recovered
– Examples: Kinetic friction, air
drag forces, normal forces,
tension forces, applied forces …
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Conservative Forces
• A force is conservative if the work it does on an
object moving between two points is
independent of the path the objects take
between the points
– The work depends only upon the initial and final
positions of the object
– Any conservative force can have a potential energy
function associated with it
Wg  PEi  PE f  mgyi  mgy f
– Work done by gravity
– Work done by spring force
1 2 1 2
Ws  PEsi  PEsf  kxi  kx f
2
2
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Nonconservative Forces
• A force is nonconservative if the work it does
on an object depends on the path taken by the
object between its final and starting points.
– The work depends upon the movement path
– For a non-conservative force, potential energy can
NOT be defined
– Work done by a nonconservative force
 
Wnc   F  d   f k d  Wotherforces
– It is generally dissipative. The dispersal
of energy takes the form of heat or sound
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6-5 Conservative and Nonconservative
Forces
Potential energy can
only be defined for
conservative forces.
6-5 Conservative and Nonconservative
Forces
Therefore, we distinguish between the work
done by conservative forces and the work done
by nonconservative forces.
We find that the work done by nonconservative
forces is equal to the total change in kinetic and
potential energies:
(6-10)
Applying Potential Energy to Problems
Practice Problem 1
By how much does the gravitational potential
energy of a 64-kg pole vaulter change if her
center of mass rises about 4.0 m during the
jump?
Applying Potential Energy to Problems
Practice Problem 2 #30 in text
A 1.60-m tall person lifts a 2.10-kg book from the
ground so it is 2.20 m above the ground. What
is the potential energy of the book relative to (a)
the ground, and (b) the top of the person’s
head? (c) How is the work done by the person
related to the answers in parts (a) and (b)?
Applying Potential Energy to Problems
A 1.60-m tall person lifts a 2.10-kg book from the ground so it is 2.20 m
above the ground. What is the potential energy of the book relative to (a)
the ground, and (b) the top of the person’s head? (c) How is the work
done by the person related to the answers in parts (a) and (b)?
(a) Relative to the ground, the PE is given by

PEG  mg  ybook  yground    2.10 kg  9.80 m s 2
  2.20 m  
45.3 J
b) Relative to the top of the person’s head, the PE is given by

PEG  mg  ybook  yhead  h   2.10 kg  9.80 m s 2
  0.60 m   12 J
c) The work done by the person in lifting the book from the ground
to the final height is the same as the answer to part (a), 45.3 J.
In part (a), the PE is calculated relative to the starting location
of the application of the force on the book. The work done by
the person is not related to the answer to part (b).
Homework
• Problems in Chapter 6
• #26, 27, 28, 29, 31, 32
Closure
• Kahoot