# Proof That The Following System Does Uniform Oscilations

But I have a question. The derivate at point ##y_0## shouldn't be:

defined as the limit of $$\frac {R(y_0 + \Delta y) - R(y_0)} {\Delta y}$$ when ## \Delta y \to 0 ## ???

You wrote:

## \frac {R(y_0 + \Delta y) - R(y_)} {\Delta y} ##
Well, we found the tension when ## y = 0 ## :

$$R(0) = kc ( a\frac{1}{ \sqrt{ c^2 - b^2 } } -1 )$$

But I have a question. The derivate at point ##y_0## shouldn't be:

defined as the limit of $$\frac {R(y_0 + \Delta y) - R(y_0)} {\Delta y}$$ when ## \Delta y \to 0 ## ???
Yes you right, I made a mistake.

$$R(0) = kc ( a\frac{1}{ \sqrt{ c^2 - b^2 } } -1 )$$
You are forgetting the force of weight in the resultant. But, again, think of this physically: ## y = 0 ## is the equilibrium point. What is the net (resultant) force on a body in equilibrium?

Ahhhhh well, it is obviously 0 . I'm sorry for not answering so quick, but personally I'm not in a hurry, and I hadn't enough time to use the laptop or my smartphone :(

Do you see then, that all that you need is to find ##R'(0)##?

So, $R(0) = kc ( a\frac{1}{ \sqrt{ c^2 - b^2 } } -1 ) + mg$

And I need to derivate this function, right ?

Yes, you need to differentiate ##R(y)##.

But you said that I need to differentiate R(0), not R(y) .

You cannot differentiate ##R(0)##. It is a number, not a function. You differentiate the function with respect to its arguments, which gives you its derivative function, then you evaluate the derivative at 0.

But I've never differentiated something like that:

$$R(y) = k(y-c)( a \frac{1}{ \sqrt{ (c-y)^2 - b^2 } } - 1 ) + mg$$

It looks difficult...

I am not sure you have to. If you have this problem while not having sufficient knowledge of calculus, it may be that you are supposed to approach it differently. I asked you earlier what you had learnt in your class before, that might give you a clue.

There is indeed a different approach. I have just talked to somebody I know, and he gave me a hint. Actually he remindend me of some basic geomtery. Well, I will shortly post the begining of the solution. You'll se that it's quite interesting, and that we both missed some things.

The solution I will post uses the aproximation of the trigonomteric functions when the angles are very small, and some basic euclidian geometry. It was fairly simple. I don't know how we both missed it...