EW-1 3/5/2009 University of Colorado at Boulder Work and Energy Energy is difficult to define because it comes in many different forms. It is hard to find a single definition which covers all the forms. Some types of energy: kinetic energy (KE) = energy of motion thermal energy = energy of "atomic jiggling" potential energy(PE) = stored energy of position/configuration various kinds of PE: gravitational electrostatic elastic (actually a form of electrostatic PE) chemical (another form of electrostatic PE) nuclear radiant energy = energy of light mass energy (Einstein's Relativity Theory says mass is a form of energy.) Almost all forms of energy on earth can be traced back to the Sun.: Example: Lift a book (gravitational PE) chemical PE in muscles chemical PE in food cows grass sun (through photosynthesis) ! Some textbooks say that energy is the ability to do work (not a bad definition, but rather abstract). A key idea that we will use over and over again is this: Whenever work is being done, energy is being changed from one form to another or being transferred from one body to another. The amount of work done on a system is the change in energy of the system. As we'll see later, energy is an extremely useful concept because energy is conserved. When we say energy is conserved, we mean that energy cannot be created or destroyed; it can only be transformed from one form to another, or transferred from one body to another. The total amount of energy everywhere is fixed; all we can do is shuffle it around. Notice that this is not what people normally mean when they say "Conserve energy." When the power company says "Conserve energy", they really mean "Don't convert the energy stored as chemical potential energy into other forms of energy too quickly." To a scientist, the phrase "conserve energy" is meaningless, because energy is always conserved. You can't NOT conserve energy. To understand energy and conservation of energy, we must first define some terms: work, kinetic energy (KE), and potential energy (PE). We’ll get to PE in the next Chapter. Let’s look at work and KE.EW-2 3/5/2009 University of Colorado at Boulder Definition of work done by a force: consider an object moving while constant forceF is applied. While the force is applied, the object moves along some axis (x-axis, say) through a displacement of magnitude | x| = d. Notice that the direction of displacement is not the same as the direction of the force, in general. Work done by a force F = F x W F d F cos d F d|| F|| = component of force along the direction of displacement, WF = F|| distance Unit of work: [W] = [F][d] = 1 N m = 1 joule = 1 J If the displacement vector is r, the work done can be written in terms of the dot product as FW F r Vector Math interlude: The dot product of two vectors A and B, "A dot B", is defined as A B ABcos The dot product of two vectors is a number, not a vector. (Later on, we will see another way to define the product of two vectors, called the "cross-product". The cross-product of two vectors is a vector.) The dot product is the magnitude of one vector (say A) times the component of the other vector (B) along the direction of the first (A). For instance, suppose that we align the x-axis with vector A. Then xA B ABcos AB. The dot product is positive, negative, or zero depending on the relative directions of the vectors A and B. When A and B are at right angles ( = 90o) , the dot product is zero. When the angle is greater then 90o, then the dot product is negative. F xi xf x| = d B B x y Bx = B cosEW-3 3/5/2009 University of Colorado at Boulder It is not difficult to prove that .. the dot product is commutative: A B B A the dot product is associative: A B C A C B C     The dot product can be written in terms of the components of the vectors like so: x x y y z zA B A B A B A B Proof (in 2D) : x y x y x x x y x x y yˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆA B A i A j B i B j A B i i A B i j ... A B A B In the last step, we used the fact that ˆ ˆ ˆ ˆi i 1 and i j=0. So, the work done by a force F is FW F r . Work is not a vector, but it does have a sign, (+) or (–). Work is positive, negative, or zero, depending on the angle between the force and the displacement. The formula WF = F d cos gives the correct sign, because cos is negative when > 90. Why do we define work this way? Answer: Whenever work is done, energy is being transformed from one form to another. The amount of work done is the amount of energy transformed. Example of work: Move book at constant velocity along a rough table with a constant horizontal force of magnitude Fext = 10 N (10 newtons). Total displacement is x = 1 m. work done by external force = WFext = + Fext x = 10 N 1 m = 10 N m = +10 J B < 90o, A B positive B = 90o, A B = 0 A > 90o, A B negative F r < 90, W positive F r = 90, W = 0 F r > 90, W negative x = +1 m Ffriction Fext = 10 NEW-4 3/5/2009 University of Colorado at Boulder Since velocity = constant, Fnet = 0, so |Fext | = |Ffric | = 10 N Work done by force of friction = WFfric = – |Ffric| | x| = – 10 J (since cos 180o = –1) Work done by normal force FN is zero: WFN = 0 (since normal force is perpendicular to displacement, cos 90o = 0.) Work done by the net force is zero. Since v = constant Fnet = 0 Wnet = 0. Moral of this example: Whenever you talk about the work done, you must be very careful to specify which force does the work. Definition of kinetic energy (KE) of an object of mass m, moving with speed v: 212KE m v KE > 0 always. An object has a big KE if it is massive and/or is moving fast. KE is energy of motion. 22units of force =[m][a]mmUnits of KE = [KE kg kg m N m J (joules)ss] Units of KE = units of work = joules Example of KE: Bowling ball (weight mg = 17 lbs, mass m = 7.7 kg ) with speed v = 7 m/s (typical bowling speed). KE = 0.5 (7.7 kg) (7 m/s)2 190 J Why do we define work and KE like we have? Because work and KE are related by …