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Nicholas J. Giordano
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Work and Energy
Introduction
• Newton’s Second Law leads to definitions of work and
energy
• The concept of energy can be applied to individual
particles or a system of particles
• The total energy of an isolated system remains constant in
time
• Known as the principle of conservation of energy
• Plays an important role in many fields
Introduction
Force, Displacement, and Work
• The connection between force and energy is work
• Work depends on the force, the displacement and the
direction between them
• Work in physics has a more specific meaning than in
everyday usage
Section 6.1
Work
• Experiments have verified that although various forces produce
different times and distances, the product of the force and the
distance remains the same
• To accelerate an object to a specific velocity, you can exert a large
force over a short distance or a small force over a long distance
Section 6.1
Work, cont.
• The product of F Δx is called
work
• For one-dimensional motion,
W = F Δx
• In two- or three-dimensions,
you must take the vector
nature of the force and
displacement into account:
W = F (Δr)cos θ
• θ is the angle between the
force and the displacement
Section 6.1
More About Work
• Units
• Newton x meter = Joule
• N. m=J
• Work is a scalar
• Although the force and displacement are both vectors
• Work can be positive or negative
• These are not directions
Section 6.1
Work and Directions
• The term F cos θ is equal to the
component of the force along
the direction of the
displacement
• When the component of the
force is parallel to the
displacement, the work is
positive
• When the component of the
force is antiparallel to the
displacement, the work is
negative
• When the component of the
force is perpendicular to the
displacement, the work is zero
Section 6.1
Relationships Among Work, Force and
Displacement
• Work is done by a force acting on an object
• The work depends on the force acting on the object and on
the object’s displacement
• The value of W depends on the direction of the force
relative to the object’s displacement
• W may be positive, negative, or zero, depending on the
angle θ between the force and the displacement
• If the displacement is zero (the object does not move),
then W = 0, even though the force may be very large
Section 6.1
Work, Physics Definition
• The term work is used in everyday language
• Its definition differs from the physics definition
• If work is positive, the object will speed up
• If work is negative, the object will slow down
Section 6.1
What Does the Work?
• When an agent applies a force to an object and does an
amount of work W on that object, the object will do an
amount of work equal to –W back on the agent
• The forces form a Newton’s Third Law action-reaction pair
• Multiple agents
• When multiple agents act on an object, you can calculate the
work done by each separate agent
Section 6.1
Graphical Analysis of Work
• So far, have assumed the
force is constant
• Look at a plot of force as a
function of the
displacement
• When the force is
constant, the graph is a
straight line
• The work is equal to the
area under the plot
Section 6.1
Graphical Analysis of Work, cont.
• The force doesn’t have to be
constant
• For each small displacement,
Δx, you can calculate the
work and then add those
results to find the total work
• The work is equal to the area
under the curve
• The area can be estimated
by dividing the area in a
series of rectangles
Section 6.1
Kinetic Energy
• Find the work done on an object as it moves from the
initial position xi to the final position xf
• W = m a Δx
• The acceleration can be expressed in terms of velocities
•
• Combining: W = ½ m vf² - ½ m vi²
• The quantity ½ m v² is called the kinetic energy
• It is the energy due to the motion of the object
Section 6.2
Work and Kinetic Energy
• The kinetic energy of an object can be changed by doing
work on the object
• W = ΔKE
• This is called the Work-Energy theorem
• The units of work and energy are the same
• Joules, J
• Another useful unit of energy is the calorie
•
1 cal = 4.186 J
Section 6.2
Work and Amplifying Force
• Suppose the person lifts
•
•
•
•
his end of the rope through
a distance L
The pulley will move
through a distance of L/2
W on crate = (2T)(L/2) = TL
W on rope = TL
Work done on the rope is
equal to the work done on
the crate
Section 6.2
Work and Amplifying Force, cont.
• The work done by the person is effectively “transferred”
to the crate
• Forces can be amplified, but work cannot be increased in
this way
• The force is amplified, but not the work
• The associated displacement is decreased
• The work-energy theorem suggests work can be converted
to energy, but since work cannot be amplified the exchange
will not increase the amount of energy available
• The result that work cannot be amplified is a consequence
of the principle of conservation of energy
Section 6.2
Potential Energy
• When an object of mass m follows any path that moves
through a vertical distance h, the work done by the
gravitational force is always equal to mgh
• W = mgh
• An object near the Earth’s surface has a potential energy
(PE) that depends only on the object’s height, h
• The PE is actually a property of the Earth-object system
Section 6.3
Potential Energy, cont.
• The work done by the
gravitational force as the
object moves from its initial
position to its final position
is independent of the path
taken
• The potential energy is
related to the work done by
the force on the object as the
object moves from one
location to another
Section 6.3
Potential Energy, final
• Relation between work and potential energy
• ΔPE = PEf – PEi = - W
• Since W is a scalar, potential energy is also a scalar
• The potential energy of an object when it is at a height y is
PE = m g y
• Applies only to objects near the Earth’s surface
• Potential energy is stored energy
• The energy can be recovered by letting the object fall back
down to its initial height, gaining kinetic energy
Section 6.3
Conservative Forces
• Conservative forces are forces that are associated with a
potential energy function
• Potential energy can be associated with forces other than
gravity
• The forces can be used to store energy as potential energy
• Forces that do not have potential energy functions
associated with them are called nonconservative forces
Section 6.3
Potential Energy and
Conservative Forces, Summary
• Potential energy is a result of the force(s) that act on an
object
• Since the forces come from the interaction between two
objects, PE is a property of the objects (the system) involved
in the force
• Potential energy is energy that an object or system has by
virtue of its position
• Potential energy is stored energy
• It can be converted to kinetic energy
Section 6.3
Potential Energy and
Conservative Forces, Summary
• Potential energy is a scalar
• Its value can be positive, negative, or zero
• It does not have a direction
• Forces can be associated with a potential energy function
• The work done is independent of the path taken
• These forces are called conservative forces
Section 6.3
Potential Energy and
Conservative Forces, Summary
• Some forces are non-conservative forces
• Examples: air drag and friction
• Do not have potential energy functions
• Cannot be used to store energy
• Work depends on the path taken by the object of interest
Adding Potential Energy to the Work-Energy
Theorem
• In the work-energy theorem (W = ΔKE), W is the work
done by all the forces acting on the object of interest
• Some of those forces can be associated with a potential
energy
• Assume all the work is done by conservative forces
• Gravity would be an example
• W = - ΔPE = ΔKE
• KEi + PEi = KEf + PEf
• Applies to all situations in which all the forces are
conservative forces
Section 6.3
Mechanical Energy
• The sum of the potential and kinetic energies is called the
mechanical energy
• Since the sum of the mechanical energy at the initial
location is equal to the sum of the mechanical energy at
the final location, the energy is conserved
• Conservation of Mechanical Energy
• KEi + PEi = KEf + PEf
• The results apply when many forces are involved as long as
they are all conservative forces
• A very powerful tool for understanding, analyzing, and
predicting motion
Section 6.3
Conservation of Energy, Example
• The snowboarder is
sliding down a frictionless
hill
• Gravity and the normal
forces are the only forces
acting on the board
• The normal is
perpendicular to the
object and so does no
work on the snowboarder
Section 6.3
Conservation of Energy, Example, cont.
• The only force that does work is gravity and it is a
conservative force
• Conservation of Mechanical Energy can be applied
• Let the initial point be the top of the hill and the final
point be the bottom of the hill
• KEi + PEi = KEf + PEf → ½ m vi² + m g yi = ½ m vf² + m g
yf
•
•
With the origin at the bottom of the hill, yi = h and yf = 0
Solve for the unknown
• In this case, vf = ?
• The final velocity depends on the height of the hill, not the
angle
Section 6.3
Charting the Energy
• A convenient way of
illustrating conservation of
energy is with a bar chart
• The kinetic and potential
energies of the snowboarder
are shown
• The sum of the energies is
the same at the start and end
• The potential energy at the
top of the hill is transformed
into kinetic energy at the
bottom of the hill
Section 6.3
Conservation of Energy or Motion Equations?
• In the snowboarder example, the hill in A is a straight
incline and motion equations could be used to solve for
the final velocity
• In B, though, the complicated hill is more realistic
• Since the slope isn’t a constant, the acceleration is not a
constant
• Motion equations could not be used
• Easily solved using energy conservation
•
The shape of the hill has no effect on the final velocity
Section 6.3
Problem Solving Strategy
• Recognize the principle
• Find the object or system whose mechanical energy is
conserved
• Sketch the problem
• Show the initial and final states of the object
• Also include a coordinate system with an origin
•
Needed to measure the potential energy
• Identify the relationships
• Find expressions for the initial and final kinetic and
potential energies
•
One or more of these may contain unknown quantities
Section 6.3
Problem Solving Strategy, cont.
• Solve
• Equate the initial mechanical energy to the final mechanical
energy
• Solve for the unknown quantities
• Check
• Consider what the answer means
• Check that the answer makes sense
• Reminder
• This approach can only be used when the mechanical energy
is conserved
Section 6.3
Conservation of Energy and
Projectile Motion
 Example, the ball is
thrown straight upward
and returns to its starting
point
 The potential energy
varies parabolically with
time
 The kinetic energy varies
as an inverted parabola
 The total energy remains
constant
Section 6.3
Changes in Potential Energy
 The figure shows two possible choices for an origin in the problem
 The change in potential energy is the same in both cases
 It is the change in potential energy that is important
 The change in potential energy does not depend on the choice of the
origin
Section 6.3
Other Potential Energy Functions
• There are potential energy functions associated with other
forces
• Examples include
• Newton’s Law of Gravitation
• Springs
Section 6.4
Gravitational Potential Energy Extended
• A more general case of a potential energy function
associated with gravity can be based on Newton’s Law of
Gravitation
• Remember, the equation for the force of gravity is
• The negative sign indicates an attractive force
• The gravitational potential energy of two objects separated
by a distance r is
• The negative sign means the potential energy is lowered as the objects
are brought closer together
Section 6.4
Gravitational Potential Energy Extended,
cont.
• The change in potential
energy is
• An example would be the
spacecraft: its potential
energy changes as its
separation from the Earth
changes
Section 6.4
Escape Speed
• The escape speed of a
satellite is the speed
needed for it to escape
from the Earth’s
gravitational pull
• Measure distances from
the center of the Earth
• rf = ∞
• Apply Conservation of
Mechanical Energy
Section 6.4
Escape Speed, cont.
• Applying conservation of mechanical energy:
• The initial distance is the radius of the Earth
• The final distance is ∞, so PEf = 0
• The final speed is 0
Section 6.4
Escape Speed, final
• For the Earth, vi = 1.1 x 104 m/s ~ 24,000 mph
• For objects fired from different planets (or the Moon), vi
depends only on the mass and radius of the planet
• In reality, air drag would have to be taken into account
when there is an atmosphere
Section 6.4
Which Gravitational Potential Energy?
• PEgrav = m g y
• The potential energy associated with the Earth’s
gravitational force for objects near its surface
• PEgrav = - G m1 m2 / r
• The potential energy associated with the gravitational force
between any two objects separated by a distance r
• This is required for problems involving motion in the solar
system
Springs and Elastic Potential Energy
• There is a potential energy
associated with springs
and other elastic objects
• When there is no force
applied to its end, the
spring is relaxed
• Not stretched or
compressed
Section 6.4
Springs and Elastic Potential Energy, cont.
• Assume you exert a force
to stretch the spring (B)
• The spring itself exerts a
force that opposes the
stretching
• You could also compress
the spring (C)
• The spring again exerts a
force back in the opposite
direction
Section 6.4
Hooke’s Law
• The force exerted by the spring has the form
Fspring = - k x
• x is the amount the end of the spring is displaced from its
equilibrium position
•
x = 0 at equilibrium
• k is called the spring constant
•
Units are N/m
• This is known as Hooke’s Law
• Applies to objects other than the spring
Section 6.4
Hooke’s Law, cont.
• Hooke’s Law is not a “law” of physics in the same sense
as Newton’s Laws
• It is an empirical relationship that experiments show works
well for springs and some other objects
• The force is a conservative force
• A potential energy function can be associated with the
force
Section 6.4
Potential Energy Stored in a Spring
• Since the force is not
constant, the work is found
by looking at the area under
the curve of the forcedisplacement curve
• Area of triangles is ½ F x
• But F = - k x
• W = - ½ k x2
• The negative sign confirms
the force and displacement
are in opposite directions
Section 6.4
Potential Energy Stored in a Spring, Summary
• From the work, an expression for the potential energy can
be found: PEspring = ½ k x2
• The force exerted by the spring always opposes the
displacement
• So Fspring can be either positive or negative
• Depends on if the spring is stretched or compressed
• The potential energy is 0 when the spring is in its relaxed
state
• The spring potential energy always increases as a spring is
either stretched or compressed
Section 6.4
Potential Energy and Force in a Spring:
Summary
Section 6.4
Potential Energy with Multiple Forces
• Conservation of Energy states KEi + PEi = KEf + PEf
• Several different forces can contribute to the potential
energy term
• For example, there might be gravity and a spring acting on
an object: PEtotal = PEgrav + PEspring
• PEtotal = m g h + ½ k x2
Section 6.4
Elastic Forces and Holding Objects
• When you hold an object,
the work done it on is zero
• Displacement is zero, so work
is zero
• However, the muscles
deform
• Muscle fibers are slipping
• Motor-like molecules in your
muscles are moving and do
work
• There is chemical energy
expended in your muscles
Section 6.4
Non-conservative Forces
• The work done by conservative forces is independent of
the path
• The work done by non-conservative forces does depend
on the path taken
• Non-conservative forces cannot be associated with a
potential energy
• Friction is an example of a non-conservative force
Section 6.5
Friction Example
• The work done by friction
in moving the block along
path B is larger than if it
moved along path A
• Other non-conservative
forces have the same
property
Section 6.5
Non-conservative Forces and the Work-Energy
Theorem
• The work in the work-energy theorem was the total work
• This work can be due to several different forces
• Wtotal = Wcon + Wnoncon
• ΔPE = -Wcon
• Then, the work-energy theorem can be restated as
+ PEi + Wnoncon = KEf + PEf
•
•
KEi
This is the general work-energy theorem with nonconservative
forces
The final mechanical energy is equal to the initial mechanical
energy plus the work done by any nonconservative forces that act
on the object
Section 6.5
Conservation of Energy, Revisited
• Suppose a system of particles or objects exerts forces on one
another as they move about
• These forces may be conservative or non-conservative
• Assume no forces from outside the system act on the system
• So total energy will be conserved
• The mechanical energy may be converted to another form of
energy such as heat, electrical, chemical, etc.
• If some agent from outside exerts a force on one of the
particles within the system, the associated amount of work will
change the total energy of the system
• The same amount of energy must be removed from the agent,
so total energy is conserved
Section 6.5
Friction Details
• For the sliding block shown,
the energy associated with
Wnoncon goes into heating up
the surface of the block and
the surface of the floor
• This increase is associated
with more movement of the
atoms
• Total energy is still
conserved
• It is not possible to return all
the energy of the atoms back
to the block’s motion
Section 6.6
Power
• Time enters into the ideas of work and energy through the
concept of power
• The average power is defined as the rate at which the
work is being done
• Unit is watts (W)
• 1 W = 1 J/s
• Sometimes expressed as horsepower
•
1 hp = 745.7 W
• Also applies to chemical and electrical processes and
devices
Section 6.7
Power and Velocity
• Power can also be expressed in terms of the velocity at
which an object is moving
• This also applies for instantaneous power and velocity: P
=Fv
• For a given power,
• The motor can exert a large force while moving slowly
• The motor can exert a small force while moving quickly
Section 6.7
Efficiency
• The efficiency, ε, of a system can be used to find the
maximum allowable force consistent with Newton’s Laws
and conservation of mechanical energy
• The efficiency can not be greater than 1
Section 6.7
Molecular Motor Example
• Myosin can be modeled as
a motor
• It moves along long
filaments of actin
molecules
• Energy source is chemical
• The maximum force is 10
pN
• Assumes an efficiency of
1
Section 6.8