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Today: (Ch. 6 & Review)
 Work and Energy
 Review (Chapter 1 to 5)
Kinetic Energy
• The force in the work equation can be found from
Newton’s Second Law
– W = F Δx = m a Δx
• The acceleration can be expressed in terms of
velocities
v 2  v i2
–
a x 
2
• 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
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
Work and 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
Potential Energy
• The work done by the
gravitational force is always
equal to mgh and is
independent of the path taken
• W = mgh
• An object near the Earth’s
surface has a potential energy
(PE) : depends on the object’s
height, h
• The potential energy is related
to the work done by the force
in moving from position 1  2
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
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
• 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)
• 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
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 gravity
– Could be any single conservative force
– W = - ΔPE = ΔKE
• KEi + PEi = KEf + PEf
• Applies to all situations in which all the forces are
conservative forces
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 mechanical 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
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
not work on the boarder
Conservation of Energy, Example, ctd.
• The only force that does work is gravity and it is a
conservative force thus 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 = 0
• Solve for the unknown
– In this case, vf = ?
– The final velocity depends on the height of the hill, not the
angle
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
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
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
Units, Vectors and Significant
figures
Forces : Newton's Three Laws
&
Balancing and Resolving Forces
in Components
Velocity and Acceleration
&
Kinematics Equations
Weight and Apparent Weight
&
Motion with Friction
Free Fall
Equilibrium
&
Incline
Projectile Motion
Uniform Circular Motion
Centripetal Force and
Acceleration
Car on Banked Road
Horizontal and Vertical Circular
Motion
Newton’s Gravitation Law
Orbital Speed and Time Period
Tomorrow: (First Exam)
 Exam in the class