TheoMech - rolling a ball down a curve

[K.J.Healey]

This isn't a homework question (im not in school) but its nearly one for a Theomech class, or maybe just dynamics. Maybe I'm just not seeing it.


Assume you have a hill that's defined by some part of a function y=f(x) (like exponentially decreasing, or a sine wave)

and you put an object on top and let it move down the curve.
assume no friction, there is some gravity force in the y direction only.

What are the independent forces of the object at a given x direction?

so I am looking for Fx(x,f(x)) and Fy(x,f(x)) so get the force given some euation describing the curve and a position.

Would I use lagrangian mechanics to pull out the accel in each direction? or what...

thanks

 

[K.J.Healey]

heres what I've come up with so far

assuming the equation f(x), the vector describing the direction of increase from the current point xo, would be :

Vi = {f(xo+dx) - f(xo), xo} (haha, I've seen that somewhere before...)

ok, so the normal force would be orthogonal to this vector.

Vn = -dx*i + (f(xo+dx)-f(xo))*j

so Fn = |Fn|*Vn
and Fn_x = -dx

wait wait wait, so this DOESNT feel a force in the X direction? I mean if take the limit and dx->0... am I doing this the wrong way?

Its basically looking at the gradient of something... why don't i remember this stuff...

 

[Buck268]

If you are referencing your x-axis perpendicular to g, then no there is no force in the x direction. Personally, I would set the axis such that x is along the average of the function of the path of decent, y being perpendicular, g will be a resultant of two vectors, Gx and Gy. I'm not really sure what your trying to do though, so...