Day 11 – June 3 – WBL 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 2 nd 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. 5.5 Conservation of Energy PC141 Intersession 2013Slide 1

Day 11 – June 3 – WBL 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 energy as: 5.5 Conservation of Energy PC141 Intersession 2013Slide 2 the total energy of a closed (i.e. isolated) system is always conserved

Day 11 – June 3 – WBL 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: 5.5 Conservation of Energy PC141 Intersession 2013Slide 3 the total energy of an open system changes by exactly the amount of net work done on the system

Day 11 – June 3 – WBL 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). 5.5 Conservation of Energy PC141 Intersession 2013Slide 4 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

Day 11 – June 3 – WBL 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. 5.5 Conservation of Energy PC141 Intersession 2013Slide 5 a force is nonconservative if the work done by it in moving an object depends on the object’s path

Day 11 – June 3 – WBL Conservation of Energy PC141 Intersession 2013Slide 6

Day 11 – June 3 – WBL 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. 5.5 Conservation of Energy PC141 Intersession 2013Slide 7

Day 11 – June 3 – WBL Conservation of Energy PC141 Intersession 2013Slide 8

Day 11 – June 3 – WBL This figure from the text (p. 163) illustrates the conservation of E for the case of a mass dropping onto a spring. 5.5 Conservation of Energy PC141 Intersession 2013Slide 9

Day 11 – June 3 – WBL The short video shown here illustrates concepts of kinetic and potential energy as they relate to half-pipe snowboarding. 5.5 Conservation of Energy PC141 Intersession 2013Slide 10 Videos are not embedded into the PPT file. You need an internet connection to view them.

Day 11 – June 3 – WBL Conservation of Energy PC141 Intersession 2013Slide 11

Day 11 – June 3 – WBL Problem #1: Work Done by a Nonconservative Force PC141 Intersession 2013Slide 12 WBL LP 5.15 If a nonconservative force acts on an object, and does work, then… A …the object’s kinetic energy is conserved B …the object’s potential energy is conserved C …the mechanical energy is conserved D …the mechanical energy is not conserved

Day 11 – June 3 – WBL Problem #2: Two Identical Stones PC141 Intersession 2013Slide 13 WBL LP 5.19 Two identical stones are thrown from the top of a tall building. Stone 1 is thrown vertically downward with an initial speed v, and stone 2 is thrown vertically upward with the same initial speed. Neglecting air resistance, which stone hits the ground with a greater speed? A Stone 1 B Stone 2 C Both will have the same speed

Day 11 – June 3 – WBL Problem #3: How Far Does it Go? PC141 Intersession 2013Slide 14 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

Day 11 – June 3 – WBL Problem #4: Bouncing Ball PC141 Intersession 2013Slide 15 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. a)What is the ball’s speed just before hitting the ground? b)What is the speed of the ball just as it leaves the ground? c)How must energy was lost and where did it go? Solution: In class

Day 11 – June 3 – WBL Power PC141 Intersession 2013Slide 16

Day 11 – June 3 – WBL Power PC141 Intersession 2013Slide 17

Day 11 – June 3 – WBL Power PC141 Intersession 2013Slide 18

Day 11 – June 3 – WBL Problem #5: Race Car PC141 Intersession 2013Slide 19 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

Day 11 – June 3 – WBL Problem #6: Construction Hoist PC141 Intersession 2013Slide 20 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

Day 11 – June 3 – WBL Problem #7: Vertical Spring PC141 Intersession 2013Slide 21 WBL Ex 5.77