Simple Nature - Light and Matter

stroke are both executed in straight lines. Since the forces are in

opposite directions, one is positive and one is negative. The cyclist’s

totalforce on the crank set is zero, but the work done isn’t zero. We

have 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 to

do integrals.) Both terms are positive; one is a positive number

multiplied by a positive number, while the other is a negative times

a negative.

This might not seem like a big deal — just remember not to use

the total force — but there are many situations where the total force

is 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 are

forces, and work is done by one atom on another. When you hold a

hot potato in your hand, the transfer of heat energy through your

skin takes place with a total force that’s extremely close to zero.

At the atomic level, atoms in your skin are interacting electrically

with atoms in the potato, but the attractions and repulsions add

up to zero total force. It’s just like the cyclist’s feet acting on the

pedals, but with zillions of forces involved instead of two. There is

no practical way to measure all the individual forces, and therefore

we can’t calculate the total energy transferred.

To summarize,

∑

Fjdxj is a correct way of calculating work,

whereFjis the individual force acting on particlej, which moves a

distance dxj. However, this is only useful if you can identify all the

individual forces and determine the distance moved at each point of

contact. For convenience, I’ll refer to this as thework theorem. (It

doesn’t have a standard name.)

There is, however, something useful we can do with the total

force. We can use it to calculate the part of the work done on an

object that consists of a change in the kinetic energy it has due to

the motion of its center of mass. The proof is essentially the same

as the proof on p. 165, except that now we don’t assume the force

is acting on a single particle, so we have to be a little more delicate.

Let the object consist of nparticles. Its total kinetic energy is

K=

∑n

j=1(1/2)mjv

2

j, but this is what we’ve already realizedcan’t

be calculated using the total force. The kinetic energy it has due to

motion of its center of mass is

Kcm=

1

2

mtotalv^2 cm.

Figure r shows some examples of the distinction betweenKcmand

168 Chapter 3 Conservation of Momentum