Speaking of Wolfram Alpha

You'd be surprised what it will answer. It knows physics too, not just math ...

Using Wolfram Alpha

It can be a bit tricky to get Wofram Alpha to understand what you want. A couple of key tricks (and I'll give examples) to make it easier:

Constraints vs. real forces

It is confusing sometimes to realize what is a real, live force, and what is not. For instance, circular motion. You draw a free-body diagram, and sum up the forces. This is one side of the equation. The other side is your constraint on the motion, namely, that for a circular path a net force of mv^2/r is required. Sometimes, we call this "centripetal force," and that is highly misleading. It is not a real force, but a fictitious force, which basically means that if you impose the requirement that the motion is circular, an specific constraint is placed on your force balance.

When you think about it, your force balance always has a constraint, namely, that the net force has to result in mass times the acceleration required to produce the observed path. For a straight line path, zero acceleration, the force balance is zero: equilibrium. For a circular path, the required acceleration is v^2/r, so if circular motion is observed then the net force divided by mass must give v^2/r - if not, then you don't have circular motion. For generic paths, it is more complex: an observed path constrains the force balance, but it depends on the speed along the path and the local radius of curvature.

The confusion is, in my opinion, largely an unfortunate artifact of history and terminology. There is no centripetal force, it is just a boundary condition on your force balance that enforces circular motion.

Anyway: here's what I wrote to one of you earlier. We'll touch on this again in class tomorrow.

In the middle of page 336, in Equation 13-12 (F_n - m*a_g = m(-(omega^2)R)), why do we have a negative sign on the right side. I feel like gravitational acceleration (weight) and centripetal acceleration, both being directed toward the center of the earth, should have the same sign. Here, if you put them on the same side, they have opposite signs.

It is a little bit confusing because the centripetal force is not a true force in its own right, but only the *result* of all other forces. The language in the text is confusing in this regard.

What it is saying is that there are two real forces acting: the normal force acting upward, and the gravitational force acting inward. Since the box follows circular motion by virtue of being on the earth's (rotating) surface, we know that the *constraint* on the force balance in magnitude is that it must sum to mv^2/r. The constraint on direction is that it must be directed radially inward to result in a circular path.

Saying that "centripetal force points toward ..." or "the centripetal acceleration points ..." is misleading. The left side of the force balance contains all the real forces; the right side contains the constraints on the motion from the known path:

(1) straight line: constraint is that forces sum to zero
(2) circle: constraint is that forces give mv^2/r to be consistent with the path, pointing radially inward.
(3) general: what we derived a while back; sum of forces relates to the rate of change of speed and the local radius of curvature of the path.

So it is OK that they have the same sign - it says that the net force balance comes out in favor of gravity, and the net force (which we call the centripetal force, misleadingly) points in the same direction ... we stay on the surface and don't fly off.

Problem set 10, number 9

Having worked it out myself last night, I realized that problem 9 is a bit more ... punishing than it really ought to be. The physics is easy, but the integral you need to solve is somewhat pathalogical.

Thus, the MASSIVE HINT, which sets up the integral for you and gives you the basic result. Note that I said "Use any means necessary to evaluate the integral required." This means you can look it up, perhaps with the Wolfram Integrator.

If you read the hint carefully, there are bonus points for solving the thing the hard way, and further bonus points for proving that it reduces to our usual expression for small heights.

Further hint: don't reinvent the wheel. What are the odds I made this problem up, and what are the odds that it comes from any number of advanced mechanics books?

Tuesday's class / Gravitation

For better or worse, there will be much mathness tomorrow. Two things that can help either before or after the fact:

• Skim Ch. 13 of your text (gravitation)
• Remind yourself about motion on generic curved paths.

The idea is that we're going to "derive" Kepler's laws of planetary motion and Newton's law of universal gravitation from some basic principles and one astronomical observation (viz., the existence of orbits described by conic sections).

In the end, you will not be responsible for the derivations, only the main results. You will see the derivations again in PH301 or PH302 (and possibly MA227). The hope is to show you that the main points of Ch. 13 can be derived with a bit more work, which will ideally help you appreciate them a bit better. Also, all that scary math at the beginning of the semester will pay off again, which is nice.

Once that is out of the way, we'll work on some of the homework problems. Thursday, we will make use of our shiny new results, and be able to show that the remainder of Ch. 13 is a bunch of special cases.

Anyway: tomorrow will be a lot of 'I derive stuff not in the book and you watch' than usual, but not without good reason.

Homework 9

All problems are due by the end of Friday (meaning, realistically, before you leave for break). With one exception, they are not too bad. Here you go.

In the end there 6 questions on angular momentum and torque, and one odd math problem that somehow found its way to your problem set. You will need to know what a geometric series is ...

The even-numbered H&R problems have the following numerical results, for reference:

• 11.14: 1.34 m/s
• 11.16: 0.25
• 11.66: 32 degrees (this is the bastard problem, FWIW)

Error propagation

A nice little post on error propagation, which may be useful for your trajectory calculations ...

Notes for Tuesday's class

Update: I just heard you went over the same stuff in Cal III about ten minutes later. Excellent! Those of you not in Cal III would probably benefit from talking with you colleagues who are ... sometimes hearing the formal mathematical version is clearer than the physicist's handwaving version.

----

I have made some notes for the material we covered in class today. It is not in the book. Only tiny parts of it will be on the test, for that matter. It is very cool though.

The notes are *marginally* more rigorous than what I did in class, and probably follow a more logical path. With the benefit of hindsight, and in the absence of time restrictions, it is much easier to explain these things. Or so it seems to me, YMMV.

I realize I have gone a bit above and beyond what many of you have been exposed to in your previous math classes, but that is in a way the point. Tuesday's lecture was to familiarize you with a few mathematical concepts (in a handwaving way) that will be very useful soon, which you will undoubtedly cover in your math classes in a more rigorous way. We just want to have the machinery in hand, so we can put it to use. The math department will make sure you cover the finer points I missed.

On Thursday, we will go over the main 'take-home' points of what we need to continue on. The details of the derivations are not as important as the main results, and that is what we will focus on from now on. Basically: don't be discouraged if Tuesday's lecture seemed very abstract and difficult, there are only a few key points you need to remember, which I will summarize on Thursday.

Anyway: keep in mind I typed up these notes up in excessive haste this 'evening' over the course of a few hours. There may be many instances of typos/errors/handwaving, and I welcome any corrections, comments, or requests for clarification. Some of you have covered these topics in math classes already; I welcome any suggestions you have for making the material more accessible.

Based on the eye-rolling in class today, I realize I have a few math majors in class (or at least a few people who paid serious attention in calculus classes). I promise not to do such unspeakable things with differentials in the future, and have tried to be marginally more rigorous in the notes. Not enough, mind you, this is where your comments can come in handy :-)

Finding the magnitude of a vector

I had a question earlier about finding the magnitude of a vector. The easiest general way to find the magnitude of a vector is to take the scalar product of the vector and itself, since the angle between a vector and itself is zero:

\vec{A}\cdot\vec{A} = |\vec{A}||\vec{A}|\cos{\theta} = |\vec{A}|^2

If you know the vector in x-y component form,

\vec{A}\cdot\vec{A} = \left(A_x\,\hat{\imath}+A_y\,\hat{\jmath}\right)\cdot\left(A_x\,\hat{\imath}+A_y\,\hat{\jmath}\right) = A_x^2 + A_y^2 = |\vec{A}|^2

It works out this way because the unit vectors are orthogonal (perpendicular) in our cartesian system:

\hat{\imath}\cdot\hat{\imath}=1, \quad \hat{\imath}\cdot\hat{\jmath}=0 

If you do it this way, it will work in all coordinate systems. Treating the vector as a little right triangle only works if you know the vector in x-y component form.