BELIEVE   ME   NOT!    - -     A   SKEPTIC's   GUIDE  

. . . )?"^11.1

This is a lot like knowing that 6 is some number n multiplied by 2 and asking what n is. We figure this out by asking ourselves the question, "What do I have to multiply by 2 to get 6?" Later on we learn to call this "division" and express the question in the form, "What is n = 6/2?" but we might just as well call it "anti-multiplication" because that is how we solve it (unless it is too hard to do in our heads and we have to resort to some complicated technology like long division).

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

. . . one.^11.2

Any introductory Calculus text will explain what an integral "means" in terms of visual pictures that the right hemisphere can handle easily: whereas the derivative of f(x) is the slope of the curve, the integral of g(x) is the area under the curve. This helps to visualize the integral as the limiting case of a summation: imagine the area under the curve of g(x) from x₀ to x being divided up into N rectangular columns of equal width $\Delta x = {1 \over N} (x - x_0)$ and height g(x_n), where $x_n = n \, \Delta x$ is the position of the $n^{\rm th}$ column. If N is a small number, then $\sum_{n=1}^N g(x_n) \, \Delta x$ is a crude approximation to the area under the smooth curve; but as N gets bigger, the columns get skinnier and the approximation becomes more and more accurate and is eventually (as $N \to \infty$) exact! This is the meaning of the integral sign:

$$\int_{x_0}^x g(x) \, dx \equiv \lim_{N \to \infty} \sum_{n=1}^N g(x_n) \, \Delta x$$

$$\hbox{\rm where} \qquad \Delta x \equiv {1 \over N} (x - x_0) \qquad \hbox{\rm and} \qquad x_n = n \, \Delta x .$$

Why do I put this nice graphical description in a footnote? Because we can understand most of the Physics applications of integrals by thinking of them as "antiderivatives" and because when we go to solve an integral we almost always do it by asking the question, "What function is this the derivative of?" which means thinking of integrals as antiderivatives. This is not a complete description of the mathematics, but it is sufficient for the purposes of this course. [See? We really do "deemphasize mathematics!"]

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

. . . itself,^11.3

This also holds for the integrals of differentials of vectors.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

. . . Newton.^11.4

It should be remembered that René Descartes and Christian Huygens formulated the LAW OF CONSERVATION OF MOMENTUM before Newton's work on Mechanics. They probably deserve to be remembered as the First Modern Conservationists!

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

. . . position^11.5

In the section on CIRCULAR MOTION we chose $\mbox{\boldmath$\vec{r}$\unboldmath }$ to denote the vector position of a particle in a circular orbit, using the centre of the circle as the origin for the $\mbox{\boldmath$\vec{r}$\unboldmath }$ vector. Here we are switching to $\mbox{\boldmath$\vec{x}$\unboldmath }$ to emphasize that the current description works equally well for any type of motion, circular or otherwise. The two notations are interchangeable, but we tend to prefer $\mbox{\boldmath$\vec{x}$\unboldmath }$ when we are talking mainly about rectilinear (straight-line) motion and $\mbox{\boldmath$\vec{r}$\unboldmath }$ when we are referring our coordinates to some centre or axis.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

. . . ball.^11.6

It is unfortunate that the conventional symbol for the weight,   $\mbox{\boldmath$\vec{W}$\unboldmath }$,  uses the same letter as the conventional symbol for the work,  W. I will try to keep this straight by referring to the weight always and only in its vector form and reserving the scalar  W  for the work. But this sort of difficulty is eventually inevitable.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

. . . point.^11.7

For now, I specifically exclude cases where the ball gets going so fast that it does get airborne at some places.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

. . . upward.^11.8

Alas, another unfortunate juxtaposition of symbols! We are using   $d\mbox{\boldmath$\vec{x}$\unboldmath }$  to describe the differential vector position change and  dy  to describe the vertical component of   $d\mbox{\boldmath$\vec{x}$\unboldmath }$. Fortunately we have no cause to talk about the horizontal component in this context, or we might wish we had used   $d\mbox{\boldmath$\vec{r}$\unboldmath }$  after all!

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

. . . spring^11.9

It is important to keep careful track of who is doing work on whom, especially in this case, because if you are careless the minus signs start jumping around and multiplying like cockroaches!

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

. . . it^11.10

It doesn't matter which - if you stretch it out you have to pull in the same direction as it moves, while if you compress it you have to push in the direction of motion, so either way the force and the displacement are in the same direction and you do positive work on the spring.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

. . . comfortable^11.11

Sadly,  x₀  is not always the same for both partners in the relationship; this is a leading cause of tension in such cases. [Doesn't this metaphor extend gracefully?]

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

. . . limit.^11.12

I suspect that such foolishness is merely an example of single-valued logic [closer = better] obsessively misapplied, rather than some more insidious psychopathology. But I could be wrong!

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

. . . zero?^11.13

The choice of a zero point for  V_g  is arbitrary, of course, just like our choice of where  h = 0. This is not a problem if we allow negative potential energies [which we do!] since it is only the change in potential energy that appears in any actual mechanics problem.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

. . . energy.^11.14

[Not quite, but you can visualize lots of little atoms wiggling and jiggling seemingly at random - that's heat, sort of.]

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

. . . origin^11.15

Note that everything we discuss in this case will be with reference to the chosen origin  O, which may be chosen arbitrarily but must then be carefully remembered!

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

. . . zero.^11.16

Remember from the chapter on VECTORS that only the perpendicular parts of two vectors contribute to the cross product. Any two parallel vectors have zero cross product. A vector crossed with itself is the simplest example.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

. . . physics.^11.17

For instance, the electrostatic force between two point charges obeys exactly the same "inverse square law" as gravitation, except with a much stronger constant of proportionality and the inclusion of both positive and negative charges. We will have lots more to do with that later on!

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

. . . still!^11.18

This is pretty boring from a Physicist's point of view, but even Physicists are grateful when bridges do not collapse.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.