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 Chapter 5
 By Dr. Steve Rapp

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 Display an ability to calculate work done by a force.
 Identify the force with the work it does.
 Understand the relationship between work and energy.
 Differentiate between work and power and correctly calculate power used.

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 Differentiate between kinetic energy and potential energy.
 Understand the Law of Conservation of Mechanical Energy.

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 Work is the product of the force exerted on an object and the distance
the object moves in the direction of the force: W = Fd
 The SI unit of work is the joule
(= Nm).
 Work is a scalar quantity.

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 The magnitude of the component of force F acting in the direction of
motion is found by W = Fd cos 0

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 Ricky is mowing the lawn with a pushtype mower. He holds the handle of the mower at an
angle of 60 ^{o} and exerts a force of 255 N on the handle as he
mows. How much work does Ricky do
if he mows a for a distance of 30 m?
 W = Fd cos 0 = 255 N (30 m) (cos 60) =
 3.83 x 10^{3 }J

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 Scalar quantity
 The work done by a force is zero when the force is perpendicular to the
displacement
 If there are multiple forces acting on an object, the total work done is
the algebraic sum of the amount of work done by each force

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 Work can be positive or negative
 Positive if the force and the displacement are in the same direction
 Negative if the force and the displacement are in the opposite
direction

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 Displacement is horizontal
 Force is vertical
 cos 90° = 0

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 Work is positive when lifting the box
 Work would be negative if lowering the box

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 Energy is the ability to do work.
 Energy can be transformed from one form to another, such as in the
equation E = mc^{2}.
 Chemical energy is transformed into mechanical energy when a car burns
gasoline.
 Can you think of other transformations?

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 Any object of mass m and speed v is defined to have a kinetic energy of
 KE = 1/2 mv^{2}.
 Workkinetic energy theorem: The net work done on an object by a net
force acting on it is equal to the change in kinetic energy of the
object.
 W_{net} = KE_{f}  KE_{i}

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 An object’s kinetic energy can also be thought of as the amount of work
the moving object could do in coming to rest
 The moving hammer has kinetic energy and can do work on the nail

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 Find the kinetic energy of a 4 kg object moving at 5 m/s.
 KE = 1/2 mv^{2} = 1/2 (4 kg)(5m/s)^{2}=50 J
 How much work must be done to accelerate a 1200 kg car from rest to a
speed of 2.0 m/s, assuming there is no friction?

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 W_{net} = KE_{f}  KE_{i}
 W_{net} = 1/2 mv_{f}^{2}  1/2 mv_{i}^{2}
 W_{net} = 1/2 (1200 kg) (2m/s)^{2} 
 1/2 (1200 kg) (0 m/s)^{2} = 2400 J

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 Is often referred to as the energy of position. Why?
 Potential energy is kind of like storing energy.
 When you wind up a toy car, energy is stored in the spring of the car.
 You had to do work in order to give the car potential energy.

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 The energy that an object has as a result of its position in space near
the Earth’s surface.
 PE = mgh where m = mass, g= acceleration due to gravity, h = height.
 A 2 kg object is raised a vertical distance of 3 m. What is its potential energy?
 PE = 2 kg (9.8m/s^{2}) (3 m) = 59 J

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 PE = mgy
 Units of Potential Energy are the same as those of Work and Kinetic
Energy

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 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

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 Is the energy stored in a spring when it is compressed or stretched.
 The amount of potential energy stored in the spring is equal to the work done in stretching
it or compressing it from its original length.
 The force required to compress or stretch a spring is proportional to
the distance the spring has been stretched or compressed from its
original length.

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 The sum total of an object’s kinetic energy and potential energy is its
total mechanical energy.
 So, we can say that: E = KE + PE.

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 If the forces are all conservative, the sum of the kinetic and potential
energies is a constant.
 So, KE_{2} + PE_{2}
= KE_{1 }+ PE_{1}

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 Spring is slowly stretched from 0 to x_{max}
 F_{applied} = F_{restoring} = kx
 W = ½kx²

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 The force F, in newtons, needed
to keep a spring stretched or compressed at a distance x, in meters, is
given by Hooke’s Law,
 F = kx where k is the spring constant (N/m).
 PE_{s} = 1/2 kx^{2}

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 A spring with a spring constant of 40 N/m is stretched a distance of .5m
from its original length. A.)What
force is required to keep it stretched this distance? B.) What is the elastic potential
energy in the stretched spring?

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 F = kx = 40 N/m (.5m) = 20 N
 PE_{s} = 1/2 k x^{2} =
 1/2 (40 N/m) (.5m)^{2 } = 5J

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 Power is the amount of work (W) done per unit time (t), so P = W/t.
 Since W = Fd and v = d/t we may write: P = Fd/t or Fv.
 Thus power can also be expressed as the product of the applied force and
the velocity of the object.
 The unit of power is the watt (w).
 1 w = J/s = N m/s = kg m^{2}/s^{3}

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 Often also interested in the rate at which the energy transfer takes
place
 Power is defined as this rate of energy transfer

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 US Customary units are generally hp
 need a conversion factor
 Can define units of work or energy in terms of units of power:
 kilowatt hours (kWh) are often used in electric bills

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 A hoist lifts an object weighing 1200 N a vertical distance of 15 m in
15 sec. What work is done if the
object rises at constant speed?
What power is developed?
 W = Fd = 1200 N (15 m) = 18,000 J
 P = W/t = 18,000 J/ 15 sec = 1200 w

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 How long will it take an 800 w motor to push a boat through the water a
distance of 1000 m with a force of 7.5 N?
 P = Fd/t t =
Fd/P
 t = 7.5 N (1000 m)/800 w = 9.4 sec

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 A 60 N force exerted on an object causes it to move with a constant
velocity of 5 m/s. What power is
used?
 P = Fv = 60 N (5 m/s) = 300 J/s =
 300 w

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 Energy can be transformed from one form to another
 Essential to the study of physics, chemistry, biology, geology,
astronomy
 Can be used in place of Newton’s laws to solve certain problems more
simply

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 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

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 When the work done by a force on an object depends only on the initial
and final positions of the object, then the force is referred to as a
conservative force. The work done
is independent of the path taken.
 Conservative forces must be forces that depend upon position, rather
than speed of the object or forces that vary with time.

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 Examples of conservative forces include:
 Gravity
 Spring force
 Electromagnetic forces

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 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.
 Examples of nonconservative forces:
 kinetic friction, air drag, propulsive forces

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 The friction force is transformed from the kinetic energy of the object
into a type of energy associated with temperature
 The objects are warmer than they were before the movement
 Internal Energy is the term used for the energy associated with an
object’s temperature

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 The blue path is shorter than the red path
 The work required is less on the blue path than on the red path
 Friction depends on the path and so is a nonconservative force

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 Friction tends to decrease the final mechanical energy of the system.
 Therefore, friction is known as dissipative force, which is a
nonconservative force.
 So, E_{initial }= E_{final} + l W_{friction }l

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 Conservation in general
 To say a physical quantity is conserved is to say that the numerical
value of the quantity remains constant
 In Conservation of Energy, the total mechanical energy remains constant
 In any isolated system of objects that interact only through
conservative forces, the total mechanical energy of the system remains
constant.

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 Define the system
 Select the location of zero gravitational potential energy
 Do not change this location while solving the problem
 Determine whether or not nonconservative forces are present
 If only conservative forces are present, apply conservation of energy
and solve for the unknown

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 Define the system
 Write expressions for the total initial and final energies
 Set the W_{nc} equal to the difference between the final and
initial total energy
 Follow the general rules for solving Conservation of Energy problems

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 The point in the body at which all the mass may be considered to be
concentrated
 When using mechanical energy, the change in potential energy is related
to the change in height of the center of mass

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 The work done by a variable force acting on an object that undergoes a
displacement is equal to the area under the graph of F versus x

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 The PE of the spring is added to both sides of the conservation of
energy equation

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 When nonconservative forces are present, the total mechanical energy of
the system is not constant
 The work done by all nonconservative forces acting on parts of a system
equals the change in the mechanical energy of the system

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 In equation form:
 The energy can either cross a boundary or the energy is transformed into
a form not yet accounted for
 Friction is an example of a nonconservative force

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 By Work
 By applying a force
 Produces a displacement of the system

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 Heat
 The process of transferring heat by collisions between molecules

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 Mechanical Waves
 a disturbance propagates through a medium
 Examples include sound, water, seismic

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 Electrical transmission
 transfer by means of electrical current

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 Electromagnetic radiation
 any form of electromagnetic waves
 Light, microwaves, radio waves

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