Virtual Work - Magnetostatics - Feynman?

[Thomas2054]

In reading Feynman's "Lectures on Physics", volume 2 I have a question and have included a scan of a small section from the book.

Feynman was a big fan of using the Principle of Virtual Work, but his explanation, as least insofar as how he used it is wanting, at least for me.

The attached pdf refers to the torque on a current carrying loop. It is largely how he uses algebraic signs that throw me. In this example you can see where he is applying the principle and says, "The principle of virtual work says that the torque is the rate of change of energy with angle, …" and then he throws in the negative sign on the expression of that thought. Then, just below he sets tau to the negative of how he previously defined it just a few lines above. I agree with the final outcome, equation (15.3), but I would not have used the negative signs in either the expression for dU or for tau.

Can someone enlighten me as to his thinking?

Thanks.

Thomas

 

[Crosson]

Work is defined as the negative of the potential energy difference in general:

$$\Delta U = - W$$

Work is equivalently defined as the integral of force over displacement, or in this case of torque over angular displacement:

$$W = \int_{\theta_i}^{\theta_f} \tau d\theta$$

 

[Thomas2054]

Double use of negative signs

Crosson,

Thanks for your reply.

I understand what you wrote. However, it is the "package" of negative signs that I am confused by. That is, Feynman has to define tau with a negative sign (where he says in the pdf, "Setting tau = -uBsin(theta)". Since the outcome (15.3) is correct both uses of the negative signs are required. I agree with the use of the first, but then the "package" requires the use of the second.

So why does he define tau as the inverse of the tau he had just defined only a few lines earlier in the pdf?

Thanks.

Thomas

 

[Crosson]

Thomas,

I see the problem you are pointing at, but I think the crossproduct is the source of the minus sign.

If the torque is equal to mu x B then by looking at the diagram I get the same negative sign as he does, in my case from the crossproduct and the right hand rule I deduce that the torque is in the negative y direction equal to -Iab sin(theta).

 

[Thomas2054]

Feynman's style

Crosson,

I agree that the cross product is the result of the negative sign on tau.

FWIW, Feynman's style is different than I have seen in any other physics book. Mostly I appreciate it, but not always. I suppose I have gotten used to a kind of consistency in those other books, which Feynman did not follow. That is, in this particular example, other books would have had tau with the negative sign in the first appearance, perhaps even explaining the fact that it is oriented in the negative y direction. At the very least, rather than saying "Setting tau to -uBsin(theta)…" as Feynman does in the second appearance, other books would have mentioned that the algebraic sign results from the cross product.

I think also that Feynman's "Lectures on Physics" volumes (there are 3) needs to be read in the context in which they were written. They were intended for new Cal Tech students. Feynman often says that he is using progressively more advanced math, "… as you get the same from the mathematics department." I suppose he is also assuming that the sophistication of the students is increasing and thus he does not spell out everything.

Of course, one comment he made early in volume 1 is always in the back of my mind. He said that he, personally, has a terrible time with algebraic signs, and so he just works things out and then goes back to put the signs into make everything right. I rather wish he had never said that!

Thomas