Suppose there is a cone, with the apex pointing down, and the top of the cone at height $h$, apex half-angle $\psi$, ball of mass $m$, and the initial velocity into the cone (completely horizontal, tangential to the top radius) $u$ .
My overall aim is to find an expression for angular displacement/velocity/acceleration, given $h, u, m, \psi$ And importantly $t$.
From physics at school, I know that the forces in 2 dimensions can be solved using the equations associated with a banked turn. In this case, the component of weight going down the slope would be $mg \cos\psi - mg\mu\sin\psi$; or $mg(\cos\psi - \mu\sin\psi)$ as the normal force on the slope is $mg\sin\psi$ and Friction $F = \mu*N$.
My question is this: firstly, am I right? Is there also a force in the direction of travel for the ball. Something like $-mg\mu\sin\psi$ ?
I want angular acceleration at $t$ first, so that I can integrate with respect to $t$ to find an expression for angular velocity and displacement.
So, we know $\alpha=$$a\over r$
Therefore I need an expression for $a$, acceleration tangential to the cone's circle, and $r$, the current radius the ball is at on the cone. Both at $t$.
Now I believe that the tangential acceleration is just simply $\mu N \over m$. But what is the overall normal force in this case?
Next, trigonometry gives us the radius, based on the half-angle $\psi$ and the distance from the bottom of the cone $z$: $r = z\times tan(\psi)$, where $z$ can also be equal to height - vertical displacement: $r = (h-Sz)\times tan(\psi)$ (Sz is vertical displacement from top).
So we now need Sz at $t$. To get that I can use force diagrams to get force, then acceleration downwards.
This is what I have so far for the force downwards $mgcos(\psi)(cos(\psi)-\mu sin(\psi))$, so the acceleration is $gcos(\psi)(cos(\psi)-\mu sin(\psi))$. But I am not accounting for the force going up the cone due to the ball's initial velocity $u$.
- What is the overall 3D vector for the force? Where $Fx$ is the force acting in tangent to the circle, (including friction) based on $u,μ,m,ψ$ and $t$. I am really struggling to comprehend the part which causes it to go up the cone, like if you rolled a ball at a ramp with velocity, it would go up the ramp and cause normal forces, and therefore friction. This is a similar problem, but the ball is not going straight up a ramp, its curving into a circle with a variable radius!
I can draw a diagram if anyone is unsure as to the visualization of my problem.
I have also asked this question in physics.stackexchange (https://physics.stackexchange.com/q/141336/62124) : I hadn't thought of asking it there (how stupid, I know) and I think that I'm more likely to find people who know this kind of thing there, but there are still valuable minds in math.stackexhange.