Presentation Transcript

1. 5.5 Conservation of Energy Much of physics is concerned with conservation laws – we’ll learn the 3 most important of these in PC141. A conservation law dictates that a particular physical quantity is conserved, i.e. it is constant in time. Conservation of energy is perhaps the most important of these laws. It dictates that the total energy of the universe is conserved. That doesn’t mean that the total kinetic or potential energy is conserved, but it does mean that we can relate changes in one to changes in the other. By keeping track of these changes, we can solve many problems in classical mechanics without having to resort to free-body diagrams and Newton’s 2nd law (and just when you were becoming comfortable with those…). Another way of stating this conservation law is that energy can never be created or destroyed.
2. 5.5 Conservation of Energy Of course, keeping track of energy changes in the whole universe when you just want to analyze a simple apparatus on the table in front of you isn’t a pleasant prospect. Therefore, it helps to define systems. A system is defined as a quantity of matter, enclosed by boundaries (either real or imaginary). For the first few problems of this chapter, we will start by defining the system under study, just to get used to the concept. A closed or isolated system (by far the most common in PC141) is one for which there is no interaction of any kind across the boundary. Since no energy can pass through this boundary, we can restate the law of conservation of total energyas: the total energy of a closed (i.e. isolated) system is always conserved
3. 5.5 Conservation of Energy On the other hand, an open system is one for which energy or matter can interact with the outside world. An example is the earth. If we imagine a fictitious boundary enclosing the earth and its atmosphere, it’s still an open system, since solar energy can enter the system, while thermal radiation can leave it. A somewhat less precise conservation law exists for open systems: the total energy of an open system changes by exactly the amount of net work done on the system
4. 5.5 Conservation of Energy Conservative and Nonconservative Forces One thing that we skipped over in chapter 4 is that forces can be classified as conservative or nonconservative: As a corollary of this statement, one can also prove that Another point (not mentioned in the text, but important at higher levels of physics) is that a conservative force can only depend on the position or configuration of a system, and not any other parameters (such as velocity or acceleration). a force is conservative if the work done by it in moving an object is independent of the object’s path; i.e. it depends only on the initial and final positions of the object. a force is conservative if the work done by it in moving an object through a round trip is zero
5. 5.5 Conservation of Energy Conservative and Nonconservative Forces cont’ On the other hand, Friction, for example, is a nonconservative force. Taking a longer path from an initial point to a final point produces more work done by friction – this is manifested as an increase heat (thermal energy). In general, a conservative force allows you to “store” energy as potential energy, while a nonconservative force does not. In fact, the concept of potential energy is only meaningful with conservative forces. a force is nonconservativeif the work done by it in moving an object depends on the object’s path
6. 5.5 Conservation of Energy Conservation of Total Mechanical Energy Having learned about kinetic energy (K) and potential energy (U), we can finally sum them together to produce total mechanical energy (E): For a conservative system, the total mechanical energy is constant. If a system evolves in some way, we can relate the initial and final kinetic and potential energies as And, since
7. 5.5 Conservation of Energy Conservation of Total Mechanical Energy cont’ Hence, we see that while kinetic and potential energies in a conservative system may change, their sum is always constant. For example, an object projected upward from rest will lose KE and gain PE on the way to the top of its trajectory, then lose PE and gain KE on the way back down. At any point in the trajectory, we can use the object’s height to find its PE, then use conservation of total mechanical energy to find its KE, from which we can find its speed. The results will match those found in the kinematic equations of chapter 2. An example is provided on p. 161 of the text.
8. 5.5 Conservation of Energy Conservation of Total Mechanical Energy cont’ Another way of expressing the conservation of total mechanical energy is shown here. From 2 slides ago, Rearranging, we get , or This equation requires a bit of an explanation, which we’ll cover in class.
9. 5.5 Conservation of Energy This figure from the text (p. 163) illustrates the conservation of E for the case of a mass dropping onto a spring.
10. 5.5 Conservation of Energy The short video shown here illustrates concepts of kinetic and potential energy as they relate to half-pipe snowboarding. Videos are not embedded into the PPT file. You need an internet connection to view them.
11. 5.5 Conservation of Energy Total Energy and Nonconservative Forces The presence of nonconservative forces such as friction does not mean that we can’t solve problems purely using energy considerations. We just need to know how to deal with the work done by the nonconservative force. Starting with the work-KE theorem, . Then, we subdivide the work into a portion done by conservative forces ( and a portion done by nonconservative forces (. But we know that the work done by conservative forces is Therefore, Then, . The work done by nonconservative forces is equal to the change in mechanical energy.
12. Problem #1: Work Done by a Nonconservative Force WBL LP 5.15
13. Problem #2: Two Identical Stones WBL LP 5.19
14. Problem #3: How Far Does it Go? WBL Ex 5.49 A 1.00-kg block (M) is on a flat frictionless surface. It is attached to a spring initially at its relaxed length (k = 50.0 N/m). A string is attached to the block, and runs over a pulley to a 450-g dangling mass (m). If the dangling mass is released from rest, how far does it fall before stopping? Solution: In class
15. Problem #4: Bouncing Ball WBL Ex 5.51 A 0.20-kg rubber ball is dropped from a height of 1.0 m above the floor and it bounces back to a height of 0.70 m. What is the ball’s speed just before hitting the ground? What is the speed of the ball just as it leaves the ground? How must energy was lost and where did it go? Solution: In class
16. 5.6 Power Ever since we started chapter 4, we’ve neglected one very important kinematic parameter – time. Quite often, we are interested not only in what quantity of work is done, but in how long it took to do the work. The time rate of doing work is called power. The average power is the total work done divided by the time required to do the work: From its definition, we see that power has SI units of Joules per second. 1 J/s is defined as 1 Watt (W)
17. 5.6 Power If the work is done by a constant force of magnitude F that acts while an object moves through a displacement of d, then (that is, it’s the product of force and the magnitude of average velocity). If the force and displacement are at an angle to each other, then Since power expresses the time rate of work, a process that is “twice as powerful” as another can do twice the work in a given time, or the same amount of work in half the time.
18. 5.6 Power Efficiency In many branches of science, we are concerned with how well energy is converted from one form to another. The mechanical efficiency(ε) denotes the useful work output compared with the energy input: It can also be expressed in terms of power: Since this isn’t an engineering course, we won’t spend too much time thinking about efficiency. However, it may pop up from time to time in assignments and exams.
19. Problem #5: Race Car WBL Ex 5.67 A race car is driven at a constant velocity of 200 km/h on a straight, level track. The power delivered to the wheels is 150 kW. What is the total resistive force on the car? Solution: In class
20. Problem #6: Construction Hoist WBL Ex 5.73 A construction hoist exerts an upward force of 500 N on an object with a mass of 50 kg. If the hoist started from rest, determine the power it expended to lift the object vertically for 10 s under these circumstances. Solution: In class
21. Problem #7: Vertical Spring WBL Ex 5.77 You hang an ideal spring of force constant k is vertically from the ceiling, and attach an object of mass m to the loose end. You then lower the mass slowly to its equilibrium position using your hand. Show that the spring’s change in length is given by Show that the work done by the spring is Show that the work done by gravity is , and explain why these two works do not sum to zero Show that the work done by your hand is and that the hand exerted an average force of half the object’s weight. Solution: In class