Chapter 6 Conservation Of Energy The Law Of Conservation Of Energy

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Chapter 6 Conservation of EnergyToday we begin with a very useful concept – Energy.We will encounter many familiar terms that now have very specific definitions in physics. Conservation of energy Work Potential PowerIn some cases, it can be argued that these terms have a physics definition that is similar to itseveryday usage.The Law of Conservation of EnergyThe total energy in the universe is unchanged by any physical process:total energy before total energy afterFrom page 187: “In ordinary language, conserving energy means trying not to waste usefulenergy resources. In the scientific meaning of conservation, energy is always conserved nomatter what happens.”Conservation of energy is one of the few universal principles of physics. No exception has everbeen found. It applies to physical, chemical, and biological systems.Also from page 187: “Some problems can be solved using either energy conservation orNewton’s second law. Usually the energy method is easier.” Using Newton’s second lawinvolves vector methods since forces are vector quantities. Much of the time, energy involvesscalar quantities, which are much easier to deal with (and more familiar). “When deciding whichof these two approaches to use to solve a problem, try using energy conservation first.”Kangaroos are mentioned at the beginning of this chapter.http://www.youtube.com/watch?v hijYSR2MFiYForms of EnergyAt the most fundamental level there are three kinds of energy1. Kinetic energy – energy due to motion2. Potential energy – energy due tointeractiona. Gravitational potential energy –interaction between the Earthand a massb. Elastic potential energy –interaction between a spring anda mass3. Rest energy – internal energy to a bodyEnergy is measured in Joules (J).6-1

Work Suppose a force F causes an object to move a distance x parallel to F x The work done by a constant force F is defined asW F xDO NOT MEMORIZE! Suppose a constant force F causes an object to move along r not parallel to F6-2

W F r cos where is the angle between F and r . MEMORIZE.Work is a scalar quantity and can be positive, negative, or zero. Positive: between 0o and 90o Negative: between 90o and 180o Zero: 90oo Usually tension and normal force do no work. The exception is (c) below.The work done by several forces can be found from the net forceWtotal W1 W2 WN Fnet r cos Work and Kinetic EnergyChoosing the x axis along Fnet, (using x r cos )Wtotal Fnet x max xWe had an equation from Chapter 26-3

v fx vix 2a x x22a x x 12 (v fx vix )22Substituting into WtotalWtotal 12 m(v fx vix )22Since the net force is in the x-direction, ay and az are both zero. Only the x-component of thevelocity changesv f vi (v fx v fy v fz ) (vix viy viz ) v fx vix2222222222andWtotal 12 m(v f vi )22The translational kinetic energy is defined asK 12 mv2The work-kinetic energy theorem isWtotal KWhile this expression is foundational to this chapter, do not memorize. We shall derive a moreuseful form.Gravitational Potential Energy (1)The weight can do work. Toss a ball up and it slows down. In our new language, its kineticenergy decreases. The kinetic energy is converted into another form of energy we callgravitational potential energy.The change in gravitational potential energy U grav WgravIn terms of position U grav mg yThis equation is true even if the object does not travel in a straight line.6-4

The gravitational potential energy isU grav mgyThe final form of our relation isWnc K UThis is it. You need to know it. We have another entry into our cause and effect table.Another useful form isWnc K U K f K i U f U i K i U i Wnc K f U fWnc is the work done by nonconservative forces. Nonconservative forces do not have a potentialenergy. A good example is friction.The mechanical energy isE K UConservation of Mechanical EnergyWhen nonconservative forces do no work, mechanical energy is conserved:Ei E f6-5

The zero of potential energy is arbitrary. Choose whatever is convenient, usually the ground.The work done by a conservative force is independent of the path taken.Problem: A 0.1-kg ball is thrown at 5 m/s from a 10 m tower. What is its speed when it is 5 mabove the ground? What is its speed when it hits the ground?Solution: Use the conservation of mechanical energy.E1 E 2U 1 K1 U 2 K 2mgy1 12 mv1 mgy2 12 mv22gy1 12 v1 gy 2 12 v 2222v 2 v1 2 g ( y1 y 2 )2 (5 m/s ) 2 2(9.8 m/s 2 (10 m 5 m) 11.1 m/sAt the bottom, y2 0, v2 14.9 m/s.Notice the direction of the throw is not mentioned. No matter which way the ball is thrown, ithas the same speed at the same height! This is very hard to prove using Newton’s second law.Gravitational Potential Energy (2)For objects far from the Earth,U GmM ErWhile this looks very different from mgy, the text (p. 203) shows they are equivalent.Example 6.8 What is the escape velocity for the Earth?Solution: Use conservation of energy. We cannot use U mgy for an object far from the earth!Ei E fU i Ki U f K f GmmE 1GmmE 122 2 mvi 2 mv frirfWhen an object escapes from the Earth, Earth’s gravity is not acting on it and rf . If theobject barely escapes the Earth, vf 0.6-6

GmmE 12 2 mvi 0rivi 2GmEriThe starting position is on the Earth’s surface and ri RE 6.36 106 m. The mass of the Earth ismE 5.97 1024 kg. The escape velocity is2GmE2(6.67 10 11 Nm2 / kg 2 )(5.97 1024 kg)vi 11,200 m/sri(6.37 106 kg)This isvi 11,200m1 mimi 7.0s 1609 msTo summarize, we developed what is usually called the work-energy theorem:Wnc K U or K i U i Wnc K f U fwhere K is the kinetic energyK 12 mv2 ,U is the potential energy (usually gravitational)U grav mgyand Wnc is the total work performed by all nonconservative forces. In many situations (but notall), mechanical energy is conserved and Wnc 0.Work done by a variable force.The advantage of energy methods is seen when dealing with a variable force. Suppose the forcechanges with displacement. An example of such a device is shown on the next page. How dowe calculate the work done?We divide the overall displacement into a series of small displacements, x. Over eachof the smaller displacements, the force is almost constant. The work done for a smalldisplacement is W Fx x6-7

The total work is the area under the curve shown above. This procedure was used to find thegravitational potential energy for an object far from the EarthU GM E mrHooke’s Law and Ideal SpringsThe force exerted by the archerincreases as the bowstring is drawnback. Robert Hooke proposed an idealspring where the force is proportionalto the displacementFx kxThe displacement of the spring from therelaxed position is x. The constant k, iscalled the spring constant. It ismeasured in N/m and it gives thestrength of the spring. The larger k is,the stiffer the spring. The minus signindicates that if the spring is stretchedto the right, the spring pulls back to theleft and vice versa.6-8

Elastic Potential EnergyAs the spring is pulled (or pushed) from its relaxed position, work is done on it. The work doneis independent of the path taken and accordingly, a potential energy can be defined. The elasticpotential energy is found to beU elastic 12 kx2Note that U 0 when x 0.PowerOften the rate of energy conversion is important. We use the term power to refer to the rate ofenergy conversion. Over an extended time, the average power isPav E tPower is measured in Joules/second or watts (W). Be careful with W for work and W for watts.Remember that work changes the mechanical energy of the system. E tF r cos t Fv cos P Problem 82. A spring gun (k 28 N/m) is used to shoot a 56-g ball horizontally. Initially thespring is compressed by 18 cm. The ball loses contact with the spring and leaves the gun whenthe spring is still compressed by 12 cm. What is the speed of the ball when it hits the ground 1.4m below the gun?Solution: This appears to be a projectile problem. It is an energy problem with two potentialenergies. Take the initial position to be at the top and the final position just before it hits theground,6-9

0 kx1221K1 U1 K 2 U 2 mgy1 12 mv2 2 kx12 022 v2 2 gy1 k ( x1 x2 ) / m22 2(9.8 m/s 2 )(1.4 m) (28 N/m)((0.18 m) 2 (0.12 m) 2 ) /( 0.056 kg) 6.04 m/sProblem 91. A 1500-kg car coasts in neutral down a 2.0º hill. The car attains a terminal speedof 20.0 m/s. (a) How much power must the engine deliver to drive the car on a level road at 20.0m/s? (b) If the maximum useful power that can be delivered by the engine is 40.0 kW, what isthe steepest hill the car can climb at 20.0 m/s?Solution: At the terminal speed, the x-component of the weight of the car is opposed by airresistance. When coasting down the hill, the free body diagram is For the x-component (along the incline) Fx maxmg sin Fair 0Fair mg sin (1500 kg)(9.8 m/s 2 ) sin 2 513 N(a)Now the car is moving along a horizontal road at constant velocity. From the FBD foundat the top of the next page: Fx ma xFmotor Fair 0Fmotor Fair6-10

The power delivered by the engine when the car moves at 20 m/s (notice the angle below is not2º!) isPmotor Fmotorv cos Fair v cos 0 (513 N)(20 m/s ) cos 0 10,300 W(b)Again a free body diagram is helpful. Climbing with constant speed, ax 0, Fx maxFmotor mg sin Fair 0Fmotor Fair mg sin The maximum power supplied by the engine is 40.0 kW. The power isP Fmotorv cos ( Fair mg sin )v cos 0 Fair v mgv sin 6-11

sin P Fair vmgv40.0 10 3 W (513 N)(20 m/s ) (1500 kg )(9.8 m/s 2 )(20 m/s ) 0.101This corresponds to a 5.8º angle.6-12

The Law of Conservation of Energy The total energy in the universe is unchanged by any physical process: total energy before total energy after From page 187: "In ordinary language, conserving energy means trying not to waste useful energy resources. In the scientific meaning of conservation, energy is always conserved no matter what .

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