Download Chapter 11 – Work In the summary for Chapter 10

Chapter 11 – Work In the summary for Chapter 10, the idea of work was introduced, = = ‐ . It was also shown that for gravity, we could associate a gravitational potential energy with the work done by the gravitational force, = ) = ). The same association was made for the work done by stretching or squeezing a spring, = + = . In the above integral, x was measured from the equilibrium position for the spring, xe = 0. More generally the spring potential energy is . The starting point for the above discussion was the one‐dimensional version of Newton’s Second Law, Fx = m ax = m dvx/dt. Let’s look at the more general situation where F and a are vectors, F = Fx i + Fy j = ma = max i + may j. The chain rule for differentiation allows us to replace ax = dvx/dt with vx dvx/dx and ay with vy dvy/dy, Fx i + Fy j = ma = mvx dvx/dx i + mvy dvy/dy j. Imagine that the force F operates on the object as it moves over a differential distance, ds = dx i + dy j. The little bit of work done by F as the object moves ds is given by the dot or scalar product of F with ds, F•ds, (Fx i + Fy j)•(dx i + dy j) = (mvx dvx/dx i + mvy dvy/dy j)•(dx i + dy j). The definition of the dot product is given below, A•B = |
|cos θ = Ax Bx + Ay By. Remember that i and j are unit vectors with length of one that are perpendicular to one another. That means, i •i = j•j = 1 and i•j = j•i = 0. With that in mind, we can simplify F•ds to get, Fx dx + Fy dy = mvx dvx + mvy dvy Now imagine integrating the above equation from point A, (xA, yA), to point B, (xB, yB) along some line. The integral on the left, ,
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, represents the work done moving the object from point A to point B. A B This is a line integral and it generally depends on the particular path traversed in going from A to B. But for a special class of forces, called conservative forces, the value of the integral is the same for all paths that connect point A to B. For these forces, it is possible to find a function U(x, y) such that, ,
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, where U is the potential energy function associated with the force F. Meanwhile the integral on the left‐hand side of the equation just gives the change in the kinetic energy of the object caused by the work done by the force F, ,
= ,
where ,
+ ,
= , with an analogous equation for . Consequently, for a conservative force, ,
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= , or, ,
= ,
. This equation represents the conservation of mechanical energy. It can only be applied to situations with conservative forces and where none of the original mechanical energy gets converted to heat, the most likely exception, or any other form of energy not included in the above equation, light and sound are two other examples. For a conservative force, the work done moving through a small distance is just Fs Δs, where Fs is the component of the force parallel to the displacement Δs. That small amount of work is just equal to the negative of the change in potential energy, Fs Δs = ‐ΔU. Solving for Fs we get Fs = ‐ΔU/Δs, or in the limit that Δs gets very small, Fs = ‐ dU/ds. In most cases this will be used when the potential energy is known as a function of x and y, U(x, y) to find the force in the x or y directions by differentiating U(x, y). For example the gravitational potential, UG = mgy, so the gravitational force is Fy = ‐ dUG/dy = ‐mg. And for a spring, US = kx2/2 and Fx = ‐ dUS/dx = ‐ kx. In situations where the mechanical energy is not conserved, the difference between the initial and final values of the mechanical energy gives the amount of energy converted to other forms of energy. All of the above applies to the object as a single chunk of mass with velocity and position. But real objects are made of atoms and molecules bound together by forces. The atoms and molecules wiggle around with internal potential and kinetic energies that are not apparent at the macroscopic level. But when the temperature of an object increases, it is an indication that the average microscopic energy of each atom has increased. The study of the internal energy of an object is the subject of Thermodynamics and/or Statistical Mechanics and is covered in PHYS 182. The major non‐conservative force dealt with this semester is friction. Friction is caused by the interaction of atoms passing over one another and is very complicated when viewed microscopically. On the other hand, we have seen that the macroscopic manifestation of friction can be well approximated by the simple equation f = μ N. When I exert a force F = μkN on a book sliding across a table at constant speed, the sum of the horizontal forces, my push and the resisting force of kinetic friction, are zero. Although there is no change in the kinetic energy of the book, I am still doing work on the book to keep it moving at a constant speed. The work I do is necessary to overcome the force of friction and is equal to μkN times the distance I slide the book. Where does that work go? It goes into increasing the energy of the atoms in the book and table or on the macroscopic level, into increasing the temperature of the book and table. Power is the rate at which work is done. A more powerful engine can accelerate a car more quickly which corresponds to increasing its kinetic energy in less time, or can climb a hill at a higher velocity which increases its gravitational potential energy. If a small amount of work dW = F dx is done in a short span of time dt, then the rate that work is done, which by definition is power, is given by dW/dt = F dx/dt. Therefore, Power = P = dW/dt = Fdx/dt = Fv. These ideas ought to be sufficient to solve the problems in Chapter 11. The key to solving energy problems is to get a good sense of the energy transformations relevant to that particular problem. What types of mechanical energy are involved in the problem? Is mechanical energy conserved or does some of the initial energy or the work done on the system get transformed into heat, and lost to the macroscopic object? If friction is involved, the mechanical energy cannot be conserved though it is possible to calculate the amount lost to thermal energy by using the “work” done in overcoming friction, average friction force times distance over which that force acts. Like most topics in the book, the best way to get a better understanding is by doing lots of problems. Also keep in mind that energy is different from force or momentum and force and momentum are also different from one another! Not the least of the difference is that energy is a scalar while force and momentum are vectors! Be very careful when thinking about physics problems and keep the differences between energy, force, and momentum in mind.