Chapter 7 Overview (Work and Energy)
Chapter 7 Overview (Work and Energy) PHYS 111
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This 5 page Class Notes was uploaded by Wilson on Wednesday October 21, 2015. The Class Notes belongs to PHYS 111 at Indiana University of Pennsylvania taught by Dr. Haija in Summer 2015. Since its upload, it has received 34 views. For similar materials see Physics I Lecture in Physics 2 at Indiana University of Pennsylvania.
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Date Created: 10/21/15
WORK AND ENERGY 71 INTRODUCTION In this unit an alternative approach based on energy considerations is followed The energy based approach is effective and very helpful because much of the formalism is cast in terms of scalars not vectors This approach starts with a definition of work done on the object by a force or forces acting on it In addition to the kinetic energy that a moving object has another energy concept called potential energy is introduced Another quantity called the total mechanical energy which is defined as the sum of an object39s kinetic and potential energies is also introduced and discussed in a variety of examples 72 WORK FN The work W done by a force F to displace an object a distance d Fig 71 is defined as Wchos p 71 where p is the angle between F and d If the force and the displacement are in the same direction Fig 72 then the angle is 0 and W becomes gFi 71 W F d cos 0 1 Note that it is only the magnitudes of both the force and displacement d that enter into Eq 71 p is the angle between the directions of the acting force and the resulting displacement The unit of work is Joules abbreviated as J 73 WORK ENERGY THEOREM Consider a force F applied to a block of mass m along the X direction displacing it a distance d Assuming that the surface is frictionless Fig 73 the force F would then be the only force acting on the block along the X axis From Newton s second law V F m a 72 But from the equation of motion the acceleration a is v2 vi Fig 73 2X a V V Substituting for a in Eq 72 above gives F m 2 0 X 2 2 V This can be rearranged to give F X In 2 l Thatis FX mV2 73 De ning the kinetic energy K E for an object of mass m moving with a velocity v as KE mv 74 Recognize that the term on the right hand side of Eq 73 represents the difference between the object39s nal and initial kinetic energies However the term on the left hand side is the work W done on the block by the force F This shows that the work done on an object by a force F through a displacement d is equal to the change in its kinetic energy during this displacement In the above argument neither the weight of the block mg nor the force of reaction FN does any work on the block as it moves along the X aXis simply because the angle between the displacement d and each of mg and FN is 90 and cos 90 0 One may rewrite Eq 73 in a new form the work done on an object by the net force acting on it is equal to the change in its kinetic energy A K E That is Wnet A K E 75a l or Wnet5mV2 75b This relation is called the workenergy theorem It establishes the equivalence between work and energy The unit of energy is the Joule J 74 CONSERVATIVE FORCES In any motion where the total work done on an object in its round trip is zero ie Wround trip 0 J the net change in the object39s kinetic energy is zero In other words the object comes back to its initial position with a velocity equal in value to that with which it was launched This property is a feature of any conservative force such as restoring spring forces Forces that do not have this property are labeled as nonconservative forces Dissipative forces are good examples of nonconservative forces 75 POTENTIAL ENERGY Since the work done on an object by a force acting on it is equal to the change in its kinetic energy it may be concluded that the object39s ability at a certain position to do work is kept undiminished as it returns to that same position if it is under the action of conservative force only This ability at a certain position being restored by the object completely may be eXplained by assigning to the object a new quantity called potential energy that in essence is an alternative notion for ability In other words as the K E of an object thrown vertically upwards decreases as it rises another quantity potential energy designated as P E increases by an amount equal to that lost by the K E and when the object reaches its highest position V 0 K E 0 its potential energy assumes a maximum value This P E is what enables the object to come back on its own While the object is falling it gains K E and when it is back to where it was launched the KB is completely regained while the potential energy it previously had gained is completely lost ThusA K E A P E 76 The negative sign is to describe the fact that as K E decreases the P E increases and vice versa From Eq 76 AKEAPE 0 or AKEPE0 77 Since the change in the sum of the K E and the P E is zero this sum does not change That is K E P E constant 78 0139 K E P K E P Ef K E P Earbitrary position 79 De ning the sum K E P E as the total mechanical energy E i e EKEPE 710 Eq 79 becomes Ei Ef E arbitrary position 711 This relation describes the conservation of the total mechanical energy of an obiect under the in uence of a conservative force or forces acting on it From the work energy theorem Eq 75 the work done by any force or several forces conservative or non conservative is W A K E 712 which for a conservative force acting on the object becomes WC A K E 713 or We AP E 714 This is an important relation stating that the work done on an object by conservative forces is not only associated but equal to a corresponding change in its potential energy with a negative sign imposed on this equality As stated earlier the gravitational force acting on an object is a good example of a conservative force As will be discussed in the next section a spring force i e Hooke s force in a spring is another example of a conservative force EXAMPLE ON POTENTIAL ENERGY GRAVITY From Eq 714 the potential energy A P E acquired or lost by an object of mass m whose position changes from an initial level to another of a height y is Wc Fy cos 180 mgy 1 mgy This potential energy is positive if y is above the reference level and negative if it is below the reference level The choice of the reference level is totally optional Its choice does not affect the solution for the kinematics of the system 76 NONCONSERVATIVE FORCES AND THE TOTAL MECHANICAL ENERGY It has been established from the work energy theorem that the net work done on an object is equal to the change in its kinetic energy see Eq 75 That is Wnet A 715 When several forces some of which may be conservative and some nonconservative act on the object the left hand side of Eq 715 can be split into two terms one labeled as We for the net work done by the conservative forces and another WNc for work done by the nonconservative forces Accordingly WC WNC A K E But since from Eq 714 P E We the above relation becomes AP E WNC AK E or WNCAKEAPEAKEPE which using Eq 79 for the sum K E P E reduces to WNC A E 7 16 or WNC Ef E1 7 17 As can be observed from the above relation the work done by the nonconservative forces is equal to the change in the object s total mechanical energy In contrast to the work done by a conservative force the work WNc done on it by nonconservative forces is equal to the change in its total mechanical energy Since WNc is always negative it is clear that Ef for such an object is always less than E This is a direct and sensible result because nonconservative forces like friction air resistance etc are dissipative forces which are a burden using up some of the object s total mechanical energy
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