Problem Solving Strategies

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Aladdin

07:38

@JohnRennie PLease ping me when you get free

2 hours later…

John Rennie

09:21

@Aladdin hi. Sorry to be so long replying. It's a Windows update weekend and I had lots of server problems to fix. I'm here now for a few hours.

Aladdin

@JohnRennie Hi

John Rennie

@Aladdin hi :-)

Ajay Mishra

09:42

Hi @JohnRennie, I have got a problem to ask, are you free?

John Rennie

@AjayMishra hi yes I'm free :-)

Guru Vishnu

@JohnRennie: Hi sir :-)

Ajay Mishra

What I thought the answer was is $\dfrac{\omega_{+} + \omega_{-}}{\omega_{+} - \omega_{-}}$

But that is wrong.

John Rennie

@AjayMishra I have no idea I'm afraid. The last time I did this stuff was 40 years ago, and while I still remember the basic mechanics more advanced stuff like coupled system long ago escaped my memory. Sorry :-(

@GuruVishnu hi :-)

Ajay Mishra

Where $\omega_+$ is higher normal mode frequency and $\omega_{-}$ lower normal mode frequency.

@JohnRennie No problem, thanks. :)

John Rennie

09:47

@AjayMishra Isn't it just half the beat period?

i.e. $\omega+ - \omega-$

Ajay Mishra

I didn't get you. (By the way, I think I have forgot to add the factor of 0.5 in my expected answer)

@JohnRennie What is $\omega_-$, I have merely given the ratio of upper and lower normal mode frequency.

John Rennie

@AjayMishra you have the two normal modes, and when you combine them you get a beat frequency equal to the difference in the frequencies of the normal modes.

The period of that beat is the time for the amplitude of the modes to change from maximum to minimum and back again, so half this period is the answer to the question.

Ajay Mishra

Yeah, I've considered that. The beat frequency would be $ \dfrac{1}{2} (\omega_+ - \omega_-)$

And the underlying wave would have frequency of $\dfrac{1}{2} (\omega_+ + \omega_-)$

I didn't get you there, the period is not asked in the question.

John Rennie

@AjayMishra the question is effectively asking for the time taken for the amplitude of the higher frequency mode to change from a maximum to a minimum.

It asks for the number of oscillations, but that's just the time divided by the period of the normal mode.

Ajay Mishra

Still, the answer is not. $\omega_+ + \omega_-$

John Rennie

09:56

Have you been given the answer?

Ajay Mishra

John Rennie

Oops, yes sorry, that's in the question.

Lets call the higher frequency mode 100, then the lower frequency mode will be 1.

Ajay Mishra

Why?

John Rennie

because the spring constants differ by a factor of 100

Ajay Mishra

$ \omega_- = \omega_n = \sqrt{\dfrac{k}{m}}$

John Rennie

10:00

Oops, yes, a factor of 10 :-)

Ajay Mishra

Still, that's not right.

John Rennie

OK, I give up. Sorry :-(

Ajay Mishra

$\omega_+ = \sqrt{w_n^2 + 2w_s^2}$

Where $\omega_n = \sqrt{\dfrac{k}{m}}, \omega_s = \sqrt{\dfrac{k'}{m}}$

@JohnRennie Okay, thanks though.

pi-π

10:40

@JohnRennie Hello

John Rennie

@pi-π hi :-)

pi-π

@JohnRennie If we have to design a program that displays the curve of functions that we have entered using C language, how can we do that?

I mean what tools do we need for this?

John Rennie

I'm busy for a few minutes. I should be free soon.

pi-π

@JohnRennie Okay. Please ping me when you are free.

pi-π

11:23

@JohnRennie Hi

John Rennie

@pi-π hi. I'm free for a few minutes. What did you want to ask?

pi-π

@JohnRennie If we have to design a program that displays the curve of functions that we have entered using C language, how can we do that?
I mean what tools do we need for this?

John Rennie

Do you mean the user types a function and your program has to display it as a graph?

pi-π

@JohnRennie Yes. Display the graph

John Rennie

What sort of things can the user type? Is there a limited list of functions? Presumably there must be some limitations as there is a potentially infinite number of functions the user could type.

@pi-π hello?

pi-π

11:30

@JohnRennie The functions are limited. We are only considering Inverse Circular Functions, Algebraic, Logarithmic, Trigonometric and exponential

And simple transformations like replacing x by x+a or x-a or by kx or by k/x ...

John Rennie

So I guess the input would be a function name, e.g. sin or exp, and a minimum and maximum value of x. Then your function would graph y(x) over the requested range.

pi-π

@JohnRennie Yes. Something like that only

John Rennie

That all seems straightforward. The only difficult bit would be figuring out how to produce the graph. You need some form of graphics package.

Are you using GNU C? Or Microsoft C? Or some other compiler?

pi-π

@JohnRennie gcc compiler

John Rennie

To be honest I don't know what graphics packages are good to use with GNU C.

pi-π

11:39

@JohnRennie Okay

John Rennie

plotutils?

Plotutils

GNU plotutils is a set of free software command-line tools and software libraries for generating 2D plot graphics based on data sets. It is used in projects such as PSPP and UMLgraph, and in many areas of academic research, and is included in many Linux distributions such as Debian and cygwin. Windows and Mac OS X versions are also available. The library provides bindings for the C and C++ languages. Its stand-alone command-line tools can generate graphs and perform numerical calculation of spline curves and systems of ordinary differential equations. Plotutils is a GNU package and is ...

If this is for college work you might find they recommend a plotting package.

Aladdin

@JohnRennie will u be back later

pi-π

@JohnRennie They should have recommended but they didn't.

John Rennie

@Aladdin I'm here for another hour, then I won't be back until tomorrow. Do you want to chat now? I don't think there is much more I can say about GCC plotting packeges.

Aladdin

Yes

Ajay Mohan

11:42

I have a question, John. I'll ask after Aladdin's finished with his topic.

John Rennie

@AjayMohan post the question now, and I'll look at it as soon as I can.

Ajay Mohan

Alright. Consider the problem of a general surface charge distribution $\sigma(\theta,\phi)$ on a sphere of radius $R$. I want to show that $\lim_{r \to R^+} \partial_r \Phi(r,\theta,\phi) - \lim_{r \to R^-} \partial_r \Phi(r,\theta,\phi) = - \frac{\sigma(\theta,\phi)}{\epsilon_0}$. This can be shown using an infinitesimal gaussian cylindrical surface covering the area of interest. But, I'm asked to show it using the Green's function solution for $\Phi$ (equation $(2)$ in the above picture).

John Rennie

@AjayMohan I have no idea sorry. I haven't done any advanced electrostatics for decades and I only remember the simple stuff now.

Ajay Mohan

12:03

Equation (4) is just the familiar electrostatic boundary condition for the perpendicular component of the electric field (Equation (2.31) in (this)[i.sstatic.net/4Uc0w.png] image).

@JohnRennie Okay.. no problem.

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