Listening to the Feynman lectures or something similar is a very good idea. But please let this only be a secondary resource and not your main resource for learning. If you don't only rely on this, but are focused on the math and problem solving, then this could be a very good thing.
Thank you very much haruspex! I was doing the entire problem with the mindset of someone analyzing an inverse-square gravitational field such as that exterior to a sphere of uniform density, although I have absolutely no idea why I was stuck in this mindset :redface: you'll have to excuse the...
Oh sorry I meant to say the opposite of what I actually said. I was thinking of the field external to the sphere of uniform density. Inside it varies linearly with the distance from the center of the sphere of uniform density.
Thank you so much mfb. If we let ##u = r_a - r_b - l## in the equation ##m(\ddot{r}_a - \ddot{r}_b) = -2k(r_a - r_b - l) + f(r_a - r_b)## then we get ##m\ddot{u}+ (2k - f)u = fl##. Wouldn't this imply that the new period is ##T = 2\pi \sqrt{\frac{m}{2k - f}}## as opposed to ##T = 2\pi...
What about Feller's excellent two volume set on probability? https://www.amazon.com/dp/0471257087/?tag=pfamazon01-20&tag=pfamazon01-20
Especially the second volume seems to have some good stuff. There might also be an introduction to measure theory in the book.
I'm not sure I follow, at least with regards to the physical system in this problem. We have a spring with two identical masses one on each end and if the system is allowed to oscillate freely then the period should be ##T = 2\pi \sqrt{\frac{m}{2k}}##.
See for example ehild's post here...
Thanks for the reply! As for the tidal forces having the same sign: if we imagine the center of mass as a hypothetical particle relative to whom both the top and bottom mass experience tidal forces then shouldn't both the top and bottom mass be accelerating away from the center of mass under...
Homework Statement
The Attempt at a Solution
The reference to exercise 1.7 is not essential. The only part exercise 1.7 that is relevant to this exercise is the following: if a long narrow tube is drilled between antipodal points on a sphere of uniform mass density and two identical...
See http://planetmath.org/proofofheineboreltheorem under the heading "The case n=1, the closed interval is compact". This contains an elementary proof of the fact that closed intervals in ##\mathbb{R}## are compact.
If you already know the Heine-Borel theorem however, then you know that...
If you have one cycle, then you can find the inverse by reversing the cycle. So if \sigma = (1 ~2~5~3), then \sigma^{-1} = (3~5~2~1)
Then if you have a more general form, then you can calculate the inverse by the formula ##(\sigma\tau)^{-1}= \tau^{-1}\sigma^{-1}##.
For example, if you have...