Hi everyone this question might take a little bit to explain
I want to model the speed of a roller-coaster carriage at the bottom of a drop, accounting for friction. I'm trying to do this defining a function $f(x)$ that when graphed out at a certain interval draws a drop.
Maybe like this for example:
Where between approximately $x= -1$ and $x = 2/5$ is how my roller-coaster drop looks.
To find the speed not accounting for friction I can use simple energy conservation equations. To find the energy lost to friction though it is a bit harder.
I need to find the normal forces times the coefficient of friction. (Coefficient of friction is just a value between $0$ and $1$ defining how rough a surface is). This is possible using derivatives and integrals. I also need to know though the forces changing the direction of the carriage. To do this I think I would have to calculate the force due to circular motion at every point on the function.
To do that though I would somehow have to connect circles to how to function is changing over time.
So for example at a point where there is little change in the function the circle radius would be very big:
And where there is a big change the radius would be very small:
Does anyone have any idea you can do this? So for a function $f(x)$ there is a corresponding function $g(x)$ that gives the radius of a circle a point $x$
In cartesian form, for a plane curve, the curvature is given by
$$\kappa(x)=\frac{|f''(x)|}{\left(1+[f'(x)]^2\right)^\frac 32}$$
then the radius of the osculating circle is given by
$$g(x)=\frac1 {\kappa(x)}=\frac{\left(1+[f'(x)]^2\right)^\frac 32}{|f''(x)|}$$