ReviewD

# ReviewD - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2004 Review D: Potential Energy and the Conservation of Mechanical Energy D.1 Conservative and Non-conservative Force . ......................................................... 2 D.1.1 Introduction. ................................................................................................ 2 D.1.2 Conservative and Non-Conservative Forces. .............................................. 3 D.2 Potential Energy. .................................................................................................. 4 D.2.1 Change in Potential Energy . ....................................................................... 4 D.2.2 Definition: Change in Potential Energy. ..................................................... 5 D.2.3 Several Conservative Forces. ...................................................................... 6 D.2.4 Change in Mechanical Energy and Conservation of Mechanical Energy . . 6 D.2.5 Non-Conservation of Mechanical Energy . ................................................. 7 D.3 Worked Examples: Calculation of the Change in Potential Energy. ................... 8 D.3.1 Example 1: Change in Gravitational Potential Energy – Uniform Gravity 8 D.3.2 Example 2: Hooke’s Law Spring-mass system . ......................................... 9 D.3.3 Example 3: Inverse Square Gravitational Force. ...................................... 10 D.4 Energy Diagrams: One-Dimensional Example: Spring Forces . ........................ 11 D-1

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Potential Energy and the Conservation of Mechanical Energy D.1 Conservative and Non-conservative Force D.1.1 Introduction A “physical system” consists of a well-defined set of bodies that are interacting by means of forces. Any bodies that lie outside the boundary of the system reside in the “surroundings”. A state of the system is a set of measurable physical quantities that completely characterize the system. Figure 1 shows this division into system, boundary, and surroundings. Figure D.1.1: system, boundary, and surroundings. Up to now we have analyzed the dynamical evolution in time of our system under the action of forces using Newton’s Laws of Motion. We shall now introduce the concept of Conservation of Energy in order to analyze the change of state of a system. Definition: Change of Energy The total change in energy of a system and its surroundings between the final state and the initial state is zero, (D.1.1) total system surroundings 0 EE E ∆= ∆ + = Our quest is then to identify experimentally every type of change of energy for all physical processes and verify that energy is conserved. Can we really play this “zero sum” game? Is there any physical content to this concept of change of energy? The answer is that experimentally we can identify all the changes in energy. One important point to keep in mind is that if we add up all the changes in energy and do not arrive at a zero sum then we have an open scientific problem: find the missing change in energy! Our first example of this type of “energy accounting” involves mechanical energy. There will be of two types of mechanical energy, kinetic energy and potential energy . Our first task is to define what we mean by the change of the potential energy of a system. D-2
D.1.2 Conservative and Non-Conservative Forces We have defined the work done by a force G F on an object which moves from an initial point 0 x to a final point f x (with displacement 0 f x xx ≡− ), as the product of the component of the force with the displacement x F x , x WFx = (D.1.2) Does the work done on the object by the force depend on the path taken by the object?

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ReviewD - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

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