Is There a Mistake in Goldstein's Mechanics on Cos Theta Definition?

[mjordan2nd]

Hello,

I have the third edition of Goldstein which I have been using to learn mechanics. I believe I have found an error in the book, however normally when I feel such things I tend to either be misreading the situation or misunderstanding the concept. I checked Professor Safko's site on Goldstein corrections but it made no mention of the error I believe I caught. Also, I am not sure what printing of the edition I have. The method for ascertaining the printing edition on the professor's site does not seem to apply to the book I have, but again it is possible that I am missing something obvious. However, since Professor Safko's site was last updated in 2010, and I very recently bought the book, I thought perhaps I have a more recent printing than those presented on Dr. Safko's site.

That said, here's the issue. Beginning on pg. 47 in my book, the first two paragraphs state:

As an example, consider a smooth solid hemisphere of radius a placed with its flat side down and fastened to the Earth whose gravitational acceleration is g. Place a small mass M at the top of the hemisphere with an infinitesimal displacement off center so the mass slides down without friction. Choose coordinates x, y, z centered on the base of the hemisphere with z vertical and the x-z plane containing the initial motion of the mass.

Let \theta be the angle from the top of the sphere to the mass. The Lagrangian is L = \frac{1}{2}M(\dot{x}^2 + \dot{y}^2 + \dot{z}^2)-mgz. The initial condition allows us to ignore the y coordinate, so the constraint is a-\sqrt{x^2-z^2}=0. Expressing the problem in terms of r^2=x^2+z^2 and x/z = cos \theta, Lagrange's equations are . . . . .

My objection to this is the book's definition of cos theta. It appears to me that based on what has been written cos theta should be z/a. Am I wrong in this? I don't think so, because the calculation directly following this seems to coincide with the way I'm thinking about it. Please let me know if I'm off base here.

Thanks.

 

[mfb]

$cos(\theta)=\frac{z}{a}$, indeed.
Probably some weird typo, if the calculation afterwards is correct.

 

[Andrew Mason]

My objection to this is the book's definition of cos theta. It appears to me that based on what has been written cos theta should be z/a. Am I wrong in this? I don't think so, because the calculation directly following this seems to coincide with the way I'm thinking about it. Please let me know if I'm off base here.

Not only is \cos\theta \ne x/z but a - \sqrt{x^2-z^2} \ne 0. In fact \sqrt{x^2-z^2} is not real for \theta<\pi/4

AM

 

[mjordan2nd]

Thanks, that clears things up for me.

Also, the - in the square root was my fault. It should have read +.