I have a parabola slide and I am looking for a function of the downhill force which only depends on the x and/or y coordinate.
I came up with a function but I am not sure if this is correct.
You can see everything on the attached picture.

Question

I have a parabola slide and I am looking for a function of the downhill force which only depends on the x and/or y coordinate.
I came up with a function but I am not sure if this is correct. 
You can see everything on the attached picture.

anonymous · Answer

Here is what I did so far.

anonymous · Answer

I assume that it is safe to assume that the parabola is y=x^2, with the tip at (0,0)
dy/dx=2x
|dw:1351340058220:dw|$$\tan^{-1}(\frac{dy}{dx})=\theta$$
$$\theta=\tan^{-1}(2x)$$
$$F=-mgsin(\theta)$$

anonymous · Answer

So you seem in the right area

anonymous · Answer

Hmmm... actually I do not know that. It is not specified so it might be better if I write: y=ax^2.

anonymous · Answer

Then $$\theta=\arctan(a2x)$$

anonymous · Answer

Yes, it just looks kind of ugly if we plug that into the equation for the force.
F=-mgsin(arctan(2ax)

anonymous · Answer

Is this really the finaly answer?
Can´t we do something to make it simpler or look better? Is there perhaps a different way to determine the force in terms of x and/or y?

anonymous · Answer

|dw:1351340440158:dw|No, its not