1. Physics is Life1 Chapter 10 Energy and its Conservation

2. Physics is Life2 Energy and its Conservation  To understand how energy is measured  To identify the various forms of energy and to note that mass is one of them.  To note that mass and energy may be transformed into other forms of mass and energy but that in all such transformations, the sum total mass and energy remains the same.  To understand the meaning and importance of the law of conservation of mass and energy, a keystone in our understanding of the physical world. OBJECTIVES

3. Physics is Life3 Introduction The motion of an object can be analyzed in terms of Newton’s three laws of motion. In that analysis, force played a central role as the quantity determining the motion. In the last chapter, we used Newton’s Laws to obtain the of motion of an object in terms of work (W =Fd). It is important to note that our study of ideal and real machines gives us a preview of the most important property of energy: When energy is transferred from one body to another or from one form to another, no energy is lost and no new energy is created.

4. Physics is Life4 DEFINING ENERGY What is Energy? Energy is the ability to do work It is measured in units of work: J,Nm, or kgm 2 /s 2 Energy is a scalar quantity

5. Physics is Life5 How we use Newton’s Laws of motion to find the relationship between work and energy? Recall the “Big Momma” formula, v f 2 = v i 2 + 2ad and substitute a=F/m and solve for Fd, we get: Fd = ½ mv f 2 – ½ mv i 2 On the right hand side are the terms that describe the energy of the system. This energy results from motion and is represented by the symbol  KE, which is called the change of kinetic energy. The process of changing the energy of a system is defined as work, and it is represented by symbol W. In our derived equation, we can say that W= Fd Substituting  KE and W into derived equation we get: W =  KE This final equation is called the work-energy theoremwork-energy theorem

6. Physics is Life6 Kinetic Energy Sample Problem (a) What is the kinetic energy of a mass of 5.0 kg moving at 4.0 m/s? (b) If the mass was accelerated from rest for a distance of 10m, what force was applied to it? Solution (a) KE= ½mv 2 = ½(5.0kg)(4.0m/s) 2 = 40J (b) Fd= ½mv f 2 - ½mv i 2 = KE F(10m) = 40J F = 4.0N

7. Physics is Life7 Kinetic Energy, and the Work-Energy Principle Sample Problem A.145 kg baseball is thrown with a speed of 25 m/s. (a) What is the kinetic energy? (b) How much work is done on the ball to make it reach this speed, if it started from rest? Solution: (a) The kinetic energy is KE = 1/2mv 2 = 1/2(.145)(25m/s 2 ) 2 = 45J (b) Since the initial KE is zero, the net work is just W net = KE 2 - KE 1 = 45 J

8. Physics is Life8 Sample Problem When the brake is applied to a car having a mass of 1000kg, its speed is reduced from 30m/s to 20m/s. (a) How much work does the brake do on the car? (b) If the brake is applied for a distance of 25 m, what force does it exert on the car? Solution (a) Fd = ½mv f 2 - ½mv i 2 = 200,000Nm - 450,000Nm= -250,000J (b) Fd= -250,000J (F)(25m) =-250,000J F= -1.0 x 10 4 N Kinetic Energy, and the Work-Energy Principle

9. Physics is Life9 Sample Problem How much work is required to accelerate a 1000kg car from 20 m/s to 30 m/s? Solution: The net work need is equal to the increase in kinetic energy: W net = KE 2 -KE 1 = 1/2(1000kg)(30m/s 2 ) 2 - 1/2(1000kg)(20m/s 2 ) 2 = 2.5 x 10 5 J Kinetic Energy, and the Work-Energy Principle

10. Physics is Life10 Collisions and Kinetic Energy In collisions between two bodies, energy is generally transferred from one body to another as well as transformed into other energy forms. From our studies of momentum, there are two types of collisions. In an elastic collision, the sum of the kinetic energies before the collision is equal to the sum of the kinetic energies after the collision. In an inelastic collision, some of the initial kinetic enery of the colliding bodies is converted into heat and other energy forms. In both cases, what can you say about the momentum of the collisions before and after the interaction?

11. Physics is Life11 Potential Energy What is Potential Energy? Potential Energy is the ability of a body to do work because of the relative position of its parts or because of its position with respect to other bodies. What is Gravitational Potential Energy ? Depends upon its distance from Earth For a body near the surface of the earth, the gravitational potential energy is given by PE= mgh The more precise definition for the gravitational potential energy of a mass m at a distance R from the center of the earth is PE= -GMm/R where G is the gravitational constant and M is the mass of the Earth. The minus sign indicates that this is the energy the body lacks to be able to move from its present position to infinity. {Note that previous equation was derived from Newton’s Law of Universal Gravitation: F = GM 1 M 2 /R 2 }

12. Physics is Life12 Gravitational Potential Energy Sample Problem What is the potential energy of an elevator having a mass of 500 kg when it is 10 m above the bottom of the its shaft? Solution PE = mgh = (500kg)(9.8m/s)(10m) = 4.9 x 10 4 J

13. Physics is Life13 Gravitational Potential Energy Sample Problem What P.E. is gained when a 100. kg object is raised 4.00 m straight up? m = 100. kg h = 4.00 m ΔP.E. = mgΔh = (100. kg)(9.81 m/s 2 )4.00 m = 3920 J

14. Physics is Life14 Gravitational Potential Energy Sample Problem What is the gravitational potential energy of a mass of 1.0 kg situated at the surface of the earth (mass = 6.0 x 10 24 kg) where the radius 6.4x10 6 m? Solution PE= -GM 1 M 2 /R PE= -(6.67 x 10-11 m 3 /kgs 2 )(6.0 x 10 24 kg)(1.0kg)/ (6.4 x 10 6 m) PE= -6.3 x 10 7 J

15. Physics is Life15 Elastic Potential Energy The force required to compress or stretch a spring is: where k is called the spring constant, and needs to be measured for each spring. The greater the value of k, the stiffer or stronger the spring.

16. Physics is Life16 Elastic Potential Energy The force increases as the spring is stretched or compressed further. We find that the potential energy (or Work) of the compressed or stretched spring, measured from its equilibrium position, can be written: Work (Elastic PE) = ½ k x 2 Note: (a) The slope of F vs. x is the spring constant. (b) The area under the curve is ½ kx. Work is Fx, thus we get W = ½ kx 2

17. Physics is Life17 Elastic Potential Energy Sample Problem (a) The spring constant spring is 200N/m. What force is needed to compress the spring a distance of 0.025? (b) What is the energy stored in the spring? Solution (a) F= kx = (200Nm)(0.025) = 5 N (b) PE = 1/2kx 2 = 1/2(200Nm)(.025) 2 = 6.3 x 10 -2 J

18. Physics is Life18 Elastic Potential Energy Sample Problem If 15. joules of energy are stored in the stretched spring, what is the value of the spring constant? P.E. = ½ kx 2 15. J = (½)K(.50 m) 2 15. J =.50 k(.25) = 120 N/m

19. Physics is Life19 Elastic Potential Energy What is wrong with this picture (PE vs. displacement for a spring)? P.E. s = 1/2(kx 2 ) PE, X direct square Plot should be curved

20. Physics is Life20 Conservative and Nonconservative Forces Potential energy can only be defined for conservative forces. What is a conservative Force?

21. Physics is Life21 Conservative and Nonconservative Forces A force is said to be conservative if the work done by or against it in moving an object is independent of the objects path.conservative For example, it takes the same work to lift an object vertically a certain height as to carry it up an incline of the same vertical height*. Forces such as gravity, for which work done does not depend on the path taken but only the initial and final positions are called conservative forces.

22. Physics is Life22 Conservative and Nonconservative Forces Demonstration/Activity A conservative force is a force with the property that the work done in moving a particle between two points is independent of the path takenforcework To demonstrate the independence of path, use a demo cart of 2-5 kg on good wheels. First pull the cart up a ramp, using a spring scale, and record the distance and force. Then pull the cart horizontally (slowly, so that the force is minimal) for the length of the ramp, and lift it vertically to the same position, with the scale showing its full weight. Compare the work done and the final position of the cart in each case.

23. Physics is Life23 Conservative and Nonconservative Forces A force is said to be nonconservative if the work done by or against it in moving an object does depend on the object’s path. Friction is an example of a nonconservative force For example, if a crate is pushed against frictional force, it would take more work (against friction) to move a crate in a semicircular path than in a straight path from point 1 to point 2. If a crate is pushed up a ramp that has friction, it is evident that the work done is not equal to a change in potential energy.

24. Physics is Life24 When lifting a book, the work that you do "against gravity" in lifting is stored (somewhere... Physicists say that it is stored "in the gravitational field" or stored "in the Earth/book system".) and is available for kinetic energy of the book once you let go. Forces that store energy in this way are called conservative forces. Gravity is a conservative force, and there are many others. Elastic (Hooke's Law) forces, electric forces, etc. are conservative forces.Elastic (Hooke's Law) forces Conservative and Nonconservative Forces (Another Explanation) When pushing a book, the work that you do "against friction" is apparently lost - it is certainly not available to the book as kinetic energy! Forces that do not store energy are called nonconservative or dissipative forces. Friction is a nonconservative force, and there are others. Any friction-type force, like air resistance, is a nonconservative force. The energy that it removes from the system is no longer available to the system for kinetic energy.

25. Physics is Life25 Mechanical Energy and Its Conservation Law of Conservation of Energy - unless work is done on or by a system, its total mechanical energy does not change

26. Physics is Life26 Mechanical Energy and Its Conservation If there are no nonconservative forces, the sum of the changes in the kinetic energy and in the potential energy is zero:  PE +  KE = 0 We can rewrite the above equation as KE 2 + PE 2 = KE 1 + PE 1 This means that (KE + PE) at some initial point 1 is equal to the (KE + PE) at any later point 2. If the kinetic energy KE increases, then the potential energy PE must decrease by an equivalent amount to compensate. Thus, the total, KE + PE, remains constant. This is called the principle of conservation of mechanical energy for conservative forces: If only conservative forces are acting, the total mechanical energy of a system neither increases or decreases in any process. It stays constant- it is conserved.

27. Physics is Life27 Problems Solving Using Conservation of Mechanical Energy In the image on the left, the total mechanical energy is: The energy buckets (right) show how the energy moves from all potential to all kinetic.

28. Physics is Life28 Problem Solving Using Conservation of Mechanical Energy Sample Problem If the original height of the stone in the previous slide is y 1 =h=3.0m, calculate the stone’s speed when it has fallen to 1.0m above the ground. Solution: Since v 1 =0 (the moment of release), y 2 = 1.0m, and g=9.8 m/s 2, then 1/2mv 2 1 + mgy 1 = 1/2mv 2 2 + mgy 2 0 + (m)(9.8m/s 2 )(3.0m) = 1/2mv 2 2 +m(9.8m/s 2 )(1.0m) The masses cancel out, and solving for v 2, we get 6.2m/s

29. Physics is Life29 Problem Solving Using Conservation of Mechanical Energy If there is no friction, the speed of a roller coaster will depend only on its height compared to its starting height.

30. Physics is Life30 Problem Solving Using Conservation of Mechanical Energy Sample Problem Assuming the height of the hill in last slide is 40m, and the roller-coaster car starts from rest at the top, calculate (a) the speed of the roller-coaster car at the bottom of the hill, and (b) at what height it will have half this speed. Take y =0 at the bottom of the hill. Solution: (a) Assume v 1 =0, y 1 =40m and y 2 =0. Then: 1/2mv 2 1 + mgy 1 = 1/2mv 2 2 + mgy 2 : 0 + (m)(9.8m/s 2 )(40m) = 1/2mv 2 2 + 0 The masses cancel out and we find that v 2 = 28m/s (b) We use the same equation, but now v2 = 14m/s (half of 28 m/s) and y2 is unknown: 1/2mv 2 1 + mgy 1 = 1/2mv 2 2 + mgy 2 0 + (m)(9.8m/s 2 )(40m) = 1/2(14m/s) 2 + (m)(9.8m/s 2 )(y 2 ) The masses cancel out and we find y 2 = 30m

31. Physics is Life31 Problem Solving Using Conservation of Mechanical Energy Conceptual Example: Two water slides at a pool are shaped differently, but have the same length and start at the same height h. Two riders, Paul and Kathleen, start from rest at the same time time on different slides. (a) which rider, Paul or Kathleen, is traveling faster at the bottom? (b) Which rider make it to the bottom first? Ignore friction (a) Each rider’s initial potential energy mgh gets transformed into kinetic energy, so the speed v at the bottom is obtained from 1/2mv 2 =mgh. The mass cancel in this equation and so the speed will be the same regardless of the mass of the rider. Since they descend the same vertical height, they will finish with the same speed. (b) Note that Kathleen is consistently at a lower elevation that Paul for the entire trip. This means that she has converted all of her potential energy to kinetic energy earlier. Kathleen get to the bottom first.

32. Physics is Life32 Other Forms of Energy; Energy Transformations and the Conservation of Energy Some other forms of energy: Electric energy, nuclear energy, thermal energy, chemical energy. Work is done when energy is transferred from one object to another. Accounting for all forms of energy, we find that the total energy neither increases nor decreases. Energy as a whole is conserved.

33. Physics is Life33 Conservation of Mass While physicists were discovering the law of conservation of energy, chemists were finding that matter obeys a similar law. They observed that in a chemical reaction, matter may change its form but the quantity of matter left at the end of the reaction always comes out to be the same as the quantity present at the beginning. Thus, the law of the conservation of mass states:conservation of mass Matter can be changed from one form to another but its mass cannot be created or destroyed.

34. Physics is Life34 Conservation of Mass and Energy Today we know that, under certain circumstances, mass and energy are interchangeable; that mass can be converted to energy and energy can be converted into mass. These two conservation laws were therefore not exactly correct as stated but had to be amended to include the possibility of mass- energy transformations. They are now parts of the following more general law: The combined quantity of mass and energy in the universe remains constant.

35. Physics is Life35 Equivalence of Mass and Energy The relativity theory of Einstein enables us to determine how much energy is equivalent to a given quantity of energy:relativity theory E = mc 2 Where is the energy obtainable from a mass m, and c is the velocity of light.

36. Physics is Life36 Equivalence of Mass and Energy Sample Problem What energy is obtained when a 1.0 kg of mass is completely converted into energy? Solution E = mc 2 = (1.0kg)(3 x 10 8 m/s) = 9.0 x 10 16 J

37. Physics is Life37 Chapter 10 (Summary) Using Newton’s Laws as well as knowing that the acceleration can be written in terms of the velocity and the distance, we already found that the work done is In addition, we defined the kinetic energy as :

38. Physics is Life38 Chapter 10 (Summary) This means that the work done is equal to the change in the kinetic energy : If the net work is positive, the kinetic energy increases. If the net work is negative, the kinetic energy decreases. Work is a scalar quantity. Because work and kinetic energy can be equated, they must have the same units: kinetic energy is measured in joules. Area under the Force vs. Distance gives Work.

39. Physics is Life39 Chapter 10 (Summary) Gravitational Potential Energy Gravitational Potential Energy is the ability of a body to do work because of the relative position of its parts or because of its position with respect to other bodies. PE = mgh = -GM 1 M 2 /R The elastic potential energy stored in a spring when it is stretched or compressed is given by: PE = ½ kx 2elastic potential energy Energy exists in many forms and may change from one form to another. Energy may also change to mass and mass may change into energy according the relationship: E = mc 2

40. Physics is Life40 Chapter 10 (Summary) In all these energy and mass changes, the law of conservation of energy and mass applies. It states that the total quantity of energy and mass in the universe remains constant. law of conservation of energy In elastic collisions, KE before the collision is equal to the KE after the collisionelastic collisions In inelastic collisions, KE before the collision is not equal to the KE after the collision.inelastic collisions

41. Physics is Life41 Chapter 10 (Summary)

42. Physics is Life42 Chapter 10 (Summary)