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Chapter 8
Conservation of Energy
Classical Mechanics
beyond the Newtonian
Formulation
Introduction: Our approach
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Conservative and non-conservative forces
Potential Energy
Mechanical Energy and its conservation
Problem solving using the Conservation of
Mechanical Energy
• Gravitational Energy and escape velocity
• Power
• Potential Energy diagrams
Conservative and Non-conservative
Forces
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Definition of conservative forces
Examples
Usefulness
And non-conservative forces?
Potential Energy
• Defining an energy of location (including
orientation) in addition to an energy of motion
– Conservative forces make this possible
– Examples
• Relating force and potential energy
– in one and more dimensions
– good intuition from potential energy graphs
(more later)
Mechanical Energy and its
Conservation
• Derivation
• Statement(s) of Conservation of Mechanical
Energy
• A different “algebraic” approach is called for
– the is calculus hidden in the forces being
conservative
• Conservation Laws and Symmetry
– Emmy Noether’s discovery
– a different mental model of energy from Physics
Problem Solving Using the
Conservation of Mechanical Energy
• “Using” as a higher level of conceptual
understanding (following Vygotsky)
– Signs that this framework is useful
– Signs that this framework will not be useful
• Changed elements in problem solving
– Diagram/sketch of before (i) and after (f) noting?
– Mechanical Energy Inventory Ledger (see) (bar
charts, too)
• Example problems (Jeopardy Exercises 1, 2, 3)
Energy Conservation with
Dissipative Forces
• What if there are non-conservative forces?
– derivation EXERCISE: follow on white boards
• Intuition from a general conservation of
energy relation
• Note on friction and work by friction
Gravitational Potential Energy and
Escape Velocity
• What is the gravitational potential energy if
Newton’s Law of gravity is needed rather than
weight = mg?
• What is meant by “escape velocity”?
• Examples EXERCISE
Power
• What is power?
• How is power commonly expressed
(algebraically)?
• Power in terms of force on object and velocity
of object
• Efficiency (with a caution)
Potential Energy Diagrams
• What are all the elements of a potential
energy diagram? EXERCISE
• What can be learned from a potential energy
diagram? EXERCISE
• What is stable and unstable equilibrium in
terms of a potential energy diagram? EXERCISE
• Examples EXERCISE
Problem solving overview
• New elements within the framework?
– Sketch showing Wnon-con, vi and vf for ∆K, and
ri and rf for ∆U
– Mechanical Energy Ledger (show)
• Problem solving diagram for this framework (show)
• Recall the multiple frameworks context (show)
the end
Mechanical Energy Inventory
Ledger
K1 + K2 + ∙∙∙
U1 + U2 + ∙∙∙
Sum (E = ∑K + ∑U)
(1) before
(initial)
(2) after
(final)
Implement Conservation of Mechanical Energy
(3)
Sum (1) = Sum (2)
________________________________ = ________________________________
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#1
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• (1/2)(3N/m)x2 = (5kg)(9.8m/s2)(3m)
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#2
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• (1/2)(3N/m)x2 = (5kg)(9.8m/s2)(3m) +
(1/2)(5kg)(1m/s)2
next
#3
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• Students make it up the equation
(contributions from around the class)
• Professor creates the question (with a sketch
of the situation)
Using Conservation of Energy
The Physical situation
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Choose/identify objects and forces
Sketch Wnon-con,vi,vf,ri,rj for relevant objects
Mechanical Energy Inventory Ledger
Implement Cons. of Mech. Energy
Mathematical representation
Problem
Solution
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Class activity: Pose a problem (situation
+ question) where the escape velocity
idea is needed.
First: small groups working on posing
problems.
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