Simple Nature - Light and Matter

stroke are both executed in straight lines. Since the forces are inopposite directions, one is positive and one is negative. The cyclist’stotal force on the crank set is zero, but the work done isn’t zero. Wehave to add the work done by each stroke, W = F 1 ∆x 1 + F 2 ∆x 2 .(I’m pretending that both forces are constant, so we don’t have todo integrals.) Both terms are positive; one is a positive numbermultiplied by a positive number, while the other is a negative timesa negative.This might not seem like a big deal — just remember not to usethe total force — but there are many situations where the total forceis all we can measure. The ultimate example is heat conduction.Heat conduction is not supposed to be counted as a form of work,since it occurs without a force. But at the atomic level, there areforces, and work is done by one atom on another. When you hold ahot potato in your hand, the transfer of heat energy through yourskin takes place with a total force that’s extremely close to zero.At the atomic level, atoms in your skin are interacting electricallywith atoms in the potato, but the attractions and repulsions addup to zero total force. It’s just like the cyclist’s feet acting on thepedals, but with zillions of forces involved instead of two. There isno practical way to measure all the individual forces, and thereforewe can’t calculate the total energy transferred.To summarize, ∑ F j dx j is a correct way of calculating work,where F j is the individual force acting on particle j, which moves adistance dx j . However, this is only useful if you can identify all theindividual forces and determine the distance moved at each point ofcontact. For convenience, I’ll refer to this as the work theorem. (Itdoesn’t have a standard name.)There is, however, something useful we can do with the totalforce. We can use it to calculate the part of the work done on anobject that consists of a change in the kinetic energy it has due tothe motion of its center of mass. The proof is essentially the sameas the proof on p. 161, except that now we don’t assume the forceis acting on a single particle, so we have to be a little more delicate.Let the object consist of n particles. Its total kinetic energy isK = ∑ nj=1 (1/2)m jvj 2 , but this is what we’ve already realized can’tbe calculated using the total force. The kinetic energy it has due tomotion of its center of mass isK cm = 1 2 m totalv 2 cm .Figure r shows some examples of the distinction between K cm and164 Chapter 3 Conservation of Momentum