11/29/2018 Physics 253

Unit 4: Conservation of Energy Definition of Work (7-1, 7-2, 7-3) Examples of Work, Definition of Energy and the link to Work (7-3, 7-4, 7-5) Potential Energy (8-1, 8-2) Problem Solving with the Conservation of Energy (8-3,8-4) Applications of Conservation of Energy (8-5,8-6, 8-7, 8-8, 8-9). 11/29/2018 Physics 253

Conservation of Energy Last lesson we introduced the concepts of work and kinetic energy. Energy is a particularly useful concept because it turns out to be conserved in a physical process once all forms are considered. This is a profound and useful observation of nature - it gives us another tool to explore and understand physical processes. But first a brief review of work and kinetic energy and then onward… 11/29/2018 Physics 253

Review: The Concept of Kinetic Energy There are a number of forms of energy, energy of motion, energy stored in a field, the energy of heat…. We start with the energy of motion or kinetic energy as it is easily related to the work done on an object. A moving object can do work on another object by applying a force over a distance. Examples: A hammer striking a nail. It hits with great force and drives the nail a distance into the wall. The engine in a bus. It applies a force to the bus over the distance traveled. Note in both cases that the force resulted in a change in the object’s final velocity. That is, the work led to motion or kinetic energy. Let’s try to be more precise: 11/29/2018 Physics 253

Review Definition of Kinetic Energy Consider an object of mass m moving initially with speed v1. Now accelerate it uniformly to speed v2 by applying a constant force F over a distance d. V1 V2 F d 11/29/2018 Physics 253

Review: The Work-Energy Principle The connection between work and changing kinetic energy is a specific example of the work-energy principle. The ingredients were only the definition of work and the 2nd Law which applies to a net force. Note: If positive work is done the kinetic energy and velocity increase. Likewise a negative work decreases kinetic energy and velocity. Kinetic energy Increases linearly with mass Varies as the square of velocity Has units of Joules. 11/29/2018 Physics 253

Conservative Forces An important categorization of forces, because conservative forces are ones for which the conservation energy applies. Definition: A force is conservative if the work done by the force on an object moving from one point to another is independent of the path taken. Those forces that do not meet this definition are non-conservative forces. 11/29/2018 Physics 253

Friction is a non-conservative force The force of friction resists motion - it is always opposite the displacement. For a given path length L the work done is W=-(Ffr)(L) The work required to drag the crate from point 1 to point 2 is then greater for the longer path length. 11/29/2018 Physics 253

Gravity is a Conservative Force Consider an object moving along an arbitrary path from a point at height y1 to a point at height y2. (Let y2-y1=h). To calculate the work done on the object by gravity we need to use the line integral introduced last lesson: 11/29/2018 Physics 253

But from the geometry q=180-f, thus cosq=cos(180-f)=-cosf But note that dy=dlcosf No dependence on path only height, a conservative force. Note that the work is negative since the force is opposite the displacement. If the object were falling it would be positive work, corresponding to an increase in velocity 11/29/2018 Physics 253

Conservative Forces Equivalent definition: A force is conservative if the net work done by the force on an object moving around a closed path is zero. Consider Fig. (a), by our previous argument the work along either path A or B is the same or W. For the round trip in Fig. (b). From Point 1 to 2 along path A the works is +W From Point 1 to 2 along path B the force will be the same independent of the direction. But the here infinitesimal in the line integral is just the opposite what we would have used to calculate the work in Fig. (a). Thus the work is –W Summing the work along the close path in Fig (b) we have W+ (-W) = 0. 11/29/2018 Physics 253

Recovering Energy Note the second definition indicates that for a conservative force if work is done on an object it can be recovered on the return to the origin. Consider throwing a ball up. Since it slows down the gravity is doing negative work When it starts falling down it speeds up and the work or energy is recovered. This isn’t the case for a non-conservative force like friction, returning to the original point takes work - nothing is returned. 11/29/2018 Physics 253

Potential Energy Potential Energy is the energy associated with the position or configuration of an object. Examples of objects that could do work An object held at a height has gravitational potential energy A compressed but latched spring has elastic potential energy When these objects are released they can deliver work and move an object -- converting their potential energy into kinetic energy There are many forms of potential energy but there is a one-to-one correspondence between conservative forces and potential energies. 11/29/2018 Physics 253

Gravitational Energy The most familiar form of P.E. We can come up with a quantitative description and then also introduce some nomenclature In order to life a brick upward a force at least equal to it’s weight must be exerted on the brick. The work done by the person is then The work done by gravity is This is path independent. 11/29/2018 Physics 253

It’s kinetic energy is then But remember, for an object in free fall the velocity after falling from height h is It’s kinetic energy is then And by the work-energy theorem the falling object can do an amount of work equal to “mgh” on any object it strikes. To say it differently, it takes “mgh” worth of work to lift an object, but the object then has the ability to do “mgh” of work on some other object. It’s fair to say that the work done lifting the object has been stored as gravitational potential energy. For example some power companies store energy from their power plants by pumping water into reservoirs for later release and use to drive turbines. 11/29/2018 Physics 253

Gravitational Potential Energy: U The change in gravitational potential energy, U, can be defined as the work done by an external force moving an object from height y1 to height y2. DU=U2-U1=Wext=mg(y2-y1) But notice that the work done by the external force is equal to the negative of the work done by the gravitational force. DU=U2-U1=-WG=mg(y2-y1) This actually shows a relationship between the change in potential energy and the potential energy associated with gravity. In fact, we can say the gravitational potential energy at any point of height y is given by U=mgy 11/29/2018 Physics 253

Some Comments on Potential Energy This is the potential energy of the mass-Earth system. It took work to pull them apart and they can do work with they fall back towards each other. We could just as well define the potential energy at any point to be U=mgh+C where C is a constant. Since we are really interested in the difference in U between two points, the constant will also cancel. It’s convenient to choose C=0 at some point, but we have to stick with it! 11/29/2018 Physics 253

Example a: The Roller Coaster A 1000kg car moves from point A to B and then C. What is U at B and C relative to point A? That is y=0 at point A. What is the change in U when the car moves from B to C? Repeat relative to point C or y=0 at point C. 11/29/2018 Physics 253

y=0 at Point C y=0 at Point A Take upward as +y UA=mgyA= UB=mgyB= (1000kg)(9.8m/s2)(10m)= 9.8x104 J UC=mgyC= (1000kg)(9.8m/s2)(-15m)= -15x104 J To go from B to C: DU=UC-UB= -15x104 J- 9.8x104 J DU=-25x104 J y=0 at Point C UA=mgyA= (1000kg)(9.8m/s2)(15m)= 15x104 J UB=mgyB= (1000kg)(9.8m/s2)(25m)= 25x104 J UC=mgyC=mg(0)=0 To go from B to C: DU=UC-UB= 0-25x104 J DU=-25x104J Same change in energy! 11/29/2018 Physics 253