Calculating Velocity and Distance for Ascending Double Cone on Rails

[franceboy]

Thank you.
The function for the left rail is y=tan(b/2)*x+tan(b/2)*d . Then we have the equations x²+z²=r(y)²
The conclusion would be x²+z²=(R-tan(a/2)*(tan(b/2)*x+tan(b/2)*d))²
I think z can be expressed by z(x)=tan(y)*x+tan(y)*d.
Back to the exercise: How I find out where the rail touches the double cone? And what will happen at A?

 

[TSny]

The conclusion would be x²+z²=(R-tan(a/2)*(tan(b/2)*x+tan(b/2)*d))²

I think this is correct. This equation gives the projection into the x-z plane of the intersection of the cone and the plane (containing the rail and parallel to the z-axis). This intersection is the purple curve in the figure. There are a couple of ways you can try to find the point of contact of the rail and the cone:

1.) Note that the point of contact is the point where z takes on a maximum value on the purple curve. So, what is $dz/dx$ at the point of contact?

or

2.) Rearrange your equation for the intersection into the form $$\frac{(x+x_0)^2}{A^2} + \frac{z^2}{B^2} = 1$$ for certain constants $A$, $B$, and $x_0$. Interpret.

 

[franceboy]

Since z takes on it`s maximal value, z`(x)=0.
z(x)=root((R-tan(a/2)*(tan(b/2)*x+tan(b/2)*d))2 - x²)
I can solve the first equation with my calculator. Then I got the real x-value which differs from the x value of the center of mass.
What is the next step?