Chapter 7 Work & Energy Classical Mechanics beyond the Newtonian Formulation

Introduction: Our approach Begin learning: new quantities & frameworks, new systems, at a new level of knowing Work and Energy; and the Work-Energy principle (or relation, theorem) Defining descriptions which are used as physics goes beyond Classical Mechanics (Comment: In QM, motion is described and explained without trajectories as in CM.)

Something new New quantities – Work, Energy, Kinetic Energy, Potential Energy, Momentum, Impulse New frameworks – Work-Energy Principle, Impulse-Momentum Principle, Conservation of Energy, Conservation of Momentum New systems – Objects with more than translational motion, or only characterized by geometry and mass New level of knowing: constructed knowing →→

Work Done by Constant Forces Exploration of Work for simple case(s) Activities to increase conceptual precision – Sketch situation corresponding to W=Fd – …corresponding to W=FdcosΘ – Work done in situation 1, 2, 3123 Clarification – Definition of net-Work on an object – Definition of net-Work by an object Start tutorial

Key points regarding Work The sign of the Work does not depend on the coordinate system. The displacement in the definition of Work is based on the point at which the force is applied. The net Work is not necessarily derived from the net Force. Assignment (do but not hand in): reflect on, make sense of above (small handouts)

Scalar Product of Two Vectors New useful mathematics involving vectors, a special case of which related W, a scalar, and F and d, vectors. General definition of scalar (“dot”) product Use in the definition of work by a constant force Examples of the scalar product EXERCISE EXERCISE

Work Done by a Varying Force Using calculus reasoning to get a general definition – break up path, define work, take limit in general case, use “integral notation” to tell the story – In simpler case, look at F vs. x graph, and the geometric representation of work General definition in terms of the F vs. x graph using language you know

Review the Idea of the Integral Summing contributions (Riemann sum idea) From the graph (area under the curve idea) Revisit multiplication with units and the geometric meaning Notation EXERCISE EXERCISE Integral as anti-derivative (Induced from a couple of examples)

Kinetic Energy and the Work-Energy Principle Tutorial discussion, continue, finish Introduce the Work-Energy Principle and the Translational Kinetic Energy definition Deriving the Work-Energy Principle from Newton’s Second Law to get procedural knowing Lab Design Activity: Measuring the energy efficiency of a winch.

Problem solving Same general format 1.Go to physical system defined by problem 2.Choose objects and forces informed by problem 3.Make use of visual representations 4.Implement a general principle to create a mathematical representation 5.Use the mathematical representation to get a solution What are the changed elements?

Problem solving Same general format 1.Go to physical system defined by problem 2.Choose objects and forces informed by problem 3.Make use of visual representations 4.Implement a general principle to create a mathematical representation 5.Use the mathematical representation to get a solution Was your answer correct?

New visual representations Diagram showing forces, displacement, and before and after velocities for relevant objects Kinetic Energy Inventory Ledger (In addition) the combined chart for multiple frameworks – a visual representation of our complex problem-solving framework requiring constructed knowing skills The Ledger

New basic principle to implement W = ∆K (and ∆U = 0) The new problem solving diagram The finished diagram for W = ∆K (see)see The learning sequence – reviewreview The multiple frameworks diagram (see)see

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#1 back next F1F1 F2F2 F1F1 F2F2 v d |F 1 | = |F 2 | = F W 1 =? W 2 =? W 1&2 =?

#2 next back F2F2 F1F1 F2F2 F1F1 dd |F 1 | = |F 2 | = F W 1 =? W 2 =? W 1&2 =?

#3 back v the object d F F W on object by wall = ?

Kinetic Energy Inventory Ledger 1.K before = ______________ 2.K after = ______________ 3.∆K (2-1) = ______________ 4.W(1→2) = ______________ 5.Express: W = ∆K The mathematical representation of the Work- Energy Principle (assuming no change in internal energy) back

Using Work Energy Principle The Physical situation Choose/identify objects and forces Diagram with F, d, v s, v f for relevant objects Kinetic Energy Inventory Ledger Implement Work Energy Principle Mathematical representation Solution Problem back

Using Work Energy Principle The Physical situation Mathematical representation Solution Problem back

Using Work Energy Principle The Physical situation Choose/identify objects and forces Mathematical representation Solution Problem back

Using Work Energy Principle The Physical situation Choose/identify objects and forces Diagram with F, d, v s, v f for relevant objects Mathematical representation Solution Problem back

Using Work Energy Principle The Physical situation Choose/identify objects and forces Diagram with F, d, v s, v f for relevant objects Kinetic Energy Inventory Ledger Mathematical representation Solution Problem back

Using Work Energy Principle The Physical situation Choose/identify objects and forces Diagram with F, d, v s, v f for relevant objects Kinetic Energy Inventory Ledger Mathematical representation Solution Problem back Implement Work Energy Principle

Using Work Energy Principle The Physical situation Choose/identify objects and forces Diagram with F, d, v s, v f for relevant objects Kinetic Energy Inventory Ledger Implement Work Energy Principle Mathematical representation Solution Problem back

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Need for a new level of knowing: constructed knowing Recall our closer look at actual knowing Received knowledge Subjective knowledge Procedural knowledge Constructed knowledge Science classes require, at a minimum, procedural knowing; that is, use of a procedure – apart from relying on experts and authority and one’s subjective opinions. Constructed knowing recognizes that there are no absolute procedures, so we are responsible for our choices. →→

Recall a common perception of problem solving at the start of class equations (from somewhere) solution problem We’ve come a long way, but have focused on basically one procedure: using Newton’s Laws. In the next half of the class, we will introduce four more procedures from four additional conceptual frameworks. They come from Newton’s Laws, but give additional insights, and have power not easily gotten from Newton’s Laws in many situations. Our problem solving will now involve choices of different frameworks (We will make a major jump in knowing!) →→

Dot product exercise F = (1,2,-3)N d = (3,-4,5)m F∙d = ? angle between F and d = ? also

Show A∙B = ABcosƟ A B B - A Ɵ Use Law of Cosines: |B-A| 2 = |A| 2 + |B| 2 -2|A||B|cos Ɵ back

Notation As an object moves along the x axis, it experiences a force in the x direction that changes with the position along the axis according to F(x) = 7N – (6N/m)x + (2N/m 2 )x 2. Write an integral expression for the work done on the object when it goes from x=1m to x=3m. back