LECTURE 30: Conservation of energy

Transcription

1 Lectures Page 1 LECTURE 30: Conservation of energy Select LEARNING OBJECTIVES: i. ii. iii. iv. Differentiate between the vector nature of momentum conservation and the scalar nature of energy conservation. Re-define the work energy equation to include the differentiation between conservative work and nonconservative work. Be able to identify and explain internal energy conversions relating to chemical potential energy. Demonstrate the ability to read a problem, and then reflect upon what information is given to be able to determine what type of physics is relevant to the problem. TEXTBOOK CHAPTERS: Giancoli (Physics Principles with Applications 7 th ) :: 6-6, 6-7, 6-8, 6-9 Knight (College Physics : A strategic approach 3 rd ) :: 10.1, 10.6 BoxSand :: Energy ( Conservation of Energy ) WARM UP: A book is dropped from rest some height above a table. What are all the energy transformations that occur within the system consisting of the book, the earth, and the table? Hint: break this into two parts: the book falling, and the book hitting the table. Believe it or not, we have covered everything we need to know about energy to be able to use an energy analysis to solve problems. For this lecture, I want to start off with a quick review of what we have done so far. Then we will look at what is known as the "energy model" and define conservation of energy. Finally we can do a few practice problems with your new found energy analysis skill. Basic review of work kinetic energy theorem We initially started off by looking at a system that was a single object, thus all the forces acting on the system were external forces. The figure below shows an example

2 The work kinetic energy theorem for the above single object system was stated as Next we looked at systems containing more than one object. We realized that unlike a FBD analysis, we needed to consider internal forces, which can come in two forms, conservative and non-conservative forces. If the internal force was conservative we can define a potential energy function which was related to the work done by the conservative force. This definition is If the internal force was non-conservative, then we defined new types of energy to deal with their physical meaning (e.g. thermal energy and chemical energy). For example, the work done by friction if it was internal to our system was defined as change in thermal energy of the system Energy model The energy model is really just a restatement of what we covered in the review from above. However, we will draw it in a nice little diagram to help illustrate our definitions. The system includes multiple objects. Conservation of energy The goal before using an energy analysis is to define a system such that conservative forces are internal to your system so that you can use the potential energy functions associated with the resulting conservative work. For example, if you see a spring or a change in height, include the spring and the Earth in your system so that you can directly write your energy equation as If there are no net external forces acting on our system, then we can get rid of the net external work term which means the energy of our system is constant (i.e. the change in energy of our system is zero). This scenario is known as conservation of energy. Lectures Page 2

3 Lectures Page 3 scenario is known as conservation of energy. The above equation is interpreted as: within our system energy can be transformed from one form to another, but the net energy is constant. While this is a nice and compact form, you must remember that it only holds true if you defined a system where there are no net external forces. I strongly suggest you always start out with the most general form as seen in the equation with big red stars and arrows around it. Why? Because this is how I mentally assess a problem; I mentally ask myself the following line of questions: Are there objects within my system that are translating (moving left right up or down), if so then include kinetic energy; are there objects within my system that are rotating, if so include rotational kinetic energy; are there objects changing height within my system, if so include gravitational potential energy; are there springs within my system, if so include spring potential energy; are there people that I defined to be within my system, if so include the work from these people as a change in chemical potential energy; are there any external forces (conservative or non-conservative), if so then include the work from those forces. I use this line of questioning to construct my left hand side of the work energy equation (the initial state), and if is on the left hand side, you must include it on the right hand side (the final state), except for the external work and the internal non-conservative forces. Energy Transformations EXAMPLE: Consider a (Earth + ball) system. The ball is dropped from rest on top of a tall building. Identify the various energy transformations occurring within this system. PRACTICE: Consider a system (Box + Table). A box with an initial velocity slides to rest on top of a table. Identify the various energy transformations occurring within this system.

4 Lectures Page 4 PRACTICE: Consider a system (Oliver Queen + Bow + Arrow + Tree). Oliver Queen pulls back an arrow with his bow. He then releases the arrow, which flies horizontally until it gets stuck in a tree. Identify the various energy transformations occurring within this system. EXAMPLE: A box starts from rest on top of a frictionless incline, slides along the surface as seen in the figure below, and makes it to the top of the other side with some non-zero final velocity. Use an energy analysis to construct an equation to find the final velocity of the box. All surfaces are frictionless except for the region of width d as labeled in the figure.

5 Lectures Page 5 PRACTICE: A block is sliding along a level frictionless surface. Its speed at A is v A. It then encounters an incline whose total vertical rise is y B. The block travels off the upper end of the incline and becomes a projectile, striking the level surface as shown in the figure below. Construct an energy equation that would let you solve for the speed of the block at location C when the block is a distance y C above the level ground. PRACTICE: Starting from rest, a block of mass m is pushed up an incline by a constant horizontal force as shown in the figure below. The block travels a distance d along the incline surface and rises a vertical distance h. The coefficient of kinetic friction between the block and the incline is µ k. Construct an equation using an energy analysis to determine the speed of the block at the top of the incline.

6 Lectures Page 6 PRACTICE: Starting from rest, a block of mass m slides down a frictionless incline where it encounters an ideal spring at the bottom as shown in the figure below. Construct an equation to determine how far the spring compresses using energy analysis. Hint: draw a final state picture. Problems for discussion: (1) Find a roommate/friend/family member and try to explain to them in your own words what conservation of energy is and what energy transformations within a system are. Perhaps use some simplified examples to help communicate your understanding.