GRE-level conceptual questions

[ejs12006]

I'm studying for the GRE Physics, and I have a few conceptual questions that have arisen while I have been attempting practice problems, but which I do not understand even after reading the provided solutions.

1) I understand that in systems with normal mode oscillations consisting of, for example, two masses connected by springs, there is always a "symmetric" normal mode wherein the masses oscillate together, as if they were one mass. In one problem we were given two out of three eigen-frequencies of such a system (consisting of two pendulums on a tube sliding on a wire), and we were asked to choose the third form a list. apparently, neither of the two given corrosponded to the symmetric mode, and there was one in the list , f=Sqrt(g/l), that did, so that was the answer. How can you tell if an eigenfrequency corrosponds to a symmetric normal-mode?

2)In one problem we are expected to know that, in the bohr model of an atom, the ioniziation energy is proportional to the reduced mass of the electron-nucleus system. Is there an intuitive line of reasoning that explains why this is the case? is it that if one thinks of the system as literally two masses orbiting each other, the force needed to keep them in orbit is somehow proportional to the kinetic energy of the system?

3)In a fourier-series problem, we were given a picture of a square wave with a period of (2*pi)/w (where w=omega, frequency). It is an odd-function, so we know it uses a sine-series. this narrowed down the possible choices to two. One of these two contained the term sin(nwt) which, the book claims, is trivially zero for all integer n's. This I find really puzzling. it would make sense of wt was always a multiple of pi, but can't it be anything? This, I am sure, has an easy explenation, perhaps having to do with the period of this specific funciton, but I really cannot see it.

Thankyou!

 

[Redbelly98]

Welcome to PF

1) I understand that in systems with normal mode oscillations consisting of, for example, two masses connected by springs, there is always a "symmetric" normal mode wherein the masses oscillate together, as if they were one mass. In one problem we were given two out of three eigen-frequencies of such a system (consisting of two pendulums on a tube sliding on a wire), and we were asked to choose the third form a list. apparently, neither of the two given corrosponded to the symmetric mode, and there was one in the list , f=Sqrt(g/l), that did, so that was the answer. How can you tell if an eigenfrequency corrosponds to a symmetric normal-mode?

That's odd that there were 3 frequencies. For a system of 2 masses, there should only be 2 (assuming just 1-d motion).

But to answer you're question, try picturing the symmetric mode ... actually I think you mean anti-symmetric, since each mass is moving in the opposite direction as the other. Anyway, if you picture that mode, each mass behaves as a single pendulum attached to a fixed point, for which you know

ω = √(g/L)​

2)In one problem we are expected to know that, in the bohr model of an atom, the ioniziation energy is proportional to the reduced mass of the electron-nucleus system. Is there an intuitive line of reasoning that explains why this is the case? is it that if one thinks of the system as literally two masses orbiting each other, the force needed to keep them in orbit is somehow proportional to the kinetic energy of the system?

Any system of two masses, with the force on each mass directed towards the other mass, can be treated mathematically as a single mass with the force directed towards a single point (a "central force"). For the math to work out, the single mass must equal the reduced mass of the two-mass system.

3)In a fourier-series problem, we were given a picture of a square wave with a period of (2*pi)/w (where w=omega, frequency). It is an odd-function, so we know it uses a sine-series. this narrowed down the possible choices to two. One of these two contained the term sin(nwt) which, the book claims, is trivially zero for all integer n's. This I find really puzzling. it would make sense of wt was always a multiple of pi, but can't it be anything? This, I am sure, has an easy explenation, perhaps having to do with the period of this specific funciton, but I really cannot see it.

It sounds like they are using "t" to mean the period, rather than any arbitrary variable time.

 

[ejs12006]

TO clarify, the normal mode problem did include three masses: each pendulum as well as the tube they are attached to, which itself can slide along a wire.

 

[Redbelly98]

Thanks, that makes sense now.