Review problems in Mechanics I

[Elwin.Martin]

Homework Statement

"A large block of mass m₁ executes horizontal SHM as it slides across a friction-less surface under the action of a spring with spring constant k. A block of mass m₂ rests upon m₁. The coefficient of friction between the two blocks is $\mu$. Assume that m₂ does not slip relative to m₁.

(c) what is the angular frequency of oscillations $\omega$ ?
(d) What is the maximum amplitude, A, that the system can have if m₂ does not slip relative to m₁? (express in terms of $\omega$

Homework Equations

F=-kx Hooke's Law
$\omega = \sqrt{\frac{k}{m}}$

The Attempt at a Solution

I'm reviewing for a transfer exam and I'm going through their old finals problems. This came up and I'm hoping I'm not seriously oversimplifying here.

c. $\omega = \sqrt{\frac{k}{m_1+m_2}}$
d. $k(A)= \mu N$ where N is the normal force from the contact of the blocks
$kA= \mu m_2 g$
$A= \frac{\mu m_2 g}{k}$
$A= \frac{\mu m_2 g}{k}$
From $\omega = \sqrt{\frac{k}{m_1+m_2}}$ , $\frac{m_2}{k} = \frac{m_2}{\left(m_1+m_2\right)\omega^2}$
$A= \frac{\mu m_2 g}{\left(m_1+m_2\right)\omega^2}$

...This comes across as odd to me, if I had a normal force which result from both masses, I'd be considerably happier. I see no reason to express the answer in terms of $\omega$ when it comes out kind of ugly. The only explanation I have is that there is some contribution to the normal force I'm neglecting or I made some careless mistake elsewhere.

Any and all help would be great! Just trying to re-learn basic mechanics ><

Edited to add an m_2 where it needed to be.

 

[BruceW]

Your working looks fine to me. I would have got the same answer as you. I'm not sure why you think you made a mistake?...

 

[Elwin.Martin]

It's a problem from famously good(and difficult) school's freshman mechanics final. . .I'm just having difficulty believing that this, and the related problems, was kind of easy.

I'll keep working on things and see if I have some difficulties.
Side question: Does anyone know how to model small oscillations in Newtonian formalism?

 

[BruceW]

What do you mean by model small oscillations? Do you mean how the motion very close to a potential minimum is simple harmonic motion?

 

[Elwin.Martin]

What do you mean by model small oscillations? Do you mean how the motion very close to a potential minimum is simple harmonic motion?

I suppose so. I only know a little bit about the topic. There are several problems I have seen which say to "calculate small oscillations" for rigid hanging bodies and such. I understand that there is a term "small oscillations" when referring to perturbations of normal modes, typically performed in the Lagrangian formalism, but I haven't formally covered very much legitimate classical mechanics.

Something like this is what I was thinking of, http://tabitha.phas.ubc.ca/wiki/index.php/Small_Oscillations_and_Perturbed_Motion

 

[BruceW]

that's a good webpage. I can recommend the book "mathematics for physicists and engineers", for normal modes (and general maths stuff, it is the best book that I've read). But there are probably lots of other textbooks that are just as good. The webpage mentions a few different kinds of approximation methods. The two that I know are the normal modes, and near the bottom of the page, it mentions the use of perturbation of the Lagrangian itself (specifically, for the problem of the anharmonic oscillator). This use of perturbation of the Lagrangian is used in quantum mechanics a lot, I think. (Or, more specifically, perturbation of the Hamiltonian operator).

But anyway, when a question is about small oscillations in classical mechanics, then usually you should use normal modes. And if the problem is 1d, with only 1 particle in a spatially varying potential, then small oscillations will always be simple harmonic motion. That webpage actually gives a pretty good explanation of why/how the method of normal modes works. (Although I think they got a negative sign in the wrong place on one line). (And their explanation doesn't go through all the mathematical details, but that would make it difficult to follow I guess).

One thing I found puzzling the first time I was working with normal modes, is "what the hell are these matrices?!" The potential energy matrix is just the coefficients of the first terms (which are non-zero and not constant) of the Taylor series of the potential energy. The elements of the kinetic energy matrix are functions of the coordinates (since we are using a general coordinate system). And the approximation is to say that these elements are approximately constant.

 

[Elwin.Martin]

that's a good webpage. I can recommend the book "mathematics for physicists and engineers", for normal modes (and general maths stuff, it is the best book that I've read). But there are probably lots of other textbooks that are just as good. The webpage mentions a few different kinds of approximation methods. The two that I know are the normal modes, and near the bottom of the page, it mentions the use of perturbation of the Lagrangian itself (specifically, for the problem of the anharmonic oscillator). This use of perturbation of the Lagrangian is used in quantum mechanics a lot, I think. (Or, more specifically, perturbation of the Hamiltonian operator).

But anyway, when a question is about small oscillations in classical mechanics, then usually you should use normal modes. And if the problem is 1d, with only 1 particle in a spatially varying potential, then small oscillations will always be simple harmonic motion. That webpage actually gives a pretty good explanation of why/how the method of normal modes works. (Although I think they got a negative sign in the wrong place on one line). (And their explanation doesn't go through all the mathematical details, but that would make it difficult to follow I guess).

One thing I found puzzling the first time I was working with normal modes, is "what the hell are these matrices?!" The potential energy matrix is just the coefficients of the first terms (which are non-zero and not constant) of the Taylor series of the potential energy. The elements of the kinetic energy matrix are functions of the coordinates (since we are using a general coordinate system). And the approximation is to say that these elements are approximately constant.

Who is the author? There is a similar book published by Springer titled
"Mathematics for Physicists and Engineers
Fundamentals and Interactive Study Guide"
Or is this it?
( http://www.springerlink.com/content/978-3-642-00173-4#section=383386&page=1 )

 

[BruceW]

Whoops, sorry. The one I have is actually called "Mathematical Methods for Physics And Engineering", By K. F. Riley, S. J. Bence, M. P. Hobson.
But there are probably lots of other good ones out there. I guess its difficult to find out which are good and which aren't so good.

This site gives some good videos of old lectures (still likely to be much the same), and lists some recommended textbooks: http://www.edforall.net/index.php/other-disciplines/mathematics
And this is on MIT's website, showing all their maths courses, with recommended textbooks for each course http://student.mit.edu/catalog/m18a.html
Looking at universities' websites for the recommended texts should give you a good idea of some of the best maths textbooks :) sneaky, I know. There are also some maths information for free on the internet, if you google free maths textbooks.