Cone Rolling on Conical Surface Dynamics

[vvvlad]

Homework Statement

A cone with mass m=3.2kg and a half-angle α=10 rolls uniformly without slipping on a round conical surface B such that its vertex O remains stationary. The center of mass of cone A is at the same level as point O and is located at a distance l=17cml from it. The axis of the cone moves with an angular velocity ω=1.0rad/s. Find the static friction force acting on cone A.

Relevant Equations

Definition of Angular Momentum of a rigid body
newtons second law

I'm trying to understand the principle of solving such problems. Do I need to take into account the angular rotation of the cone around its generatrix along with the angular rotation around the vertical axis? My solution:
∑F = ma
∑M = dL/dt
L = Iw
v = wR (where R is the distance from point O to the center of mass)
Take the moment of force for the cone's axis of rotation as Mz = [ l ; F_friction ]
Then express Lz = [ l ; p ], where p = mv = mwr = mwl
Lz = mwl^2
Since the cone rotates uniformly, L = const ???
Mz = d(mwl^2)/dt

 

[Lnewqban]

Welcome!
I would consider cone A as a wheel of mass m rolling over cone B keeping radius l and rotating at constant angular velocity.
The problem asks only about the minimum tangential and radial force components induced by friction in order to avoid skidding of our wheel over cone B.

 

[wrobel]

A motion of a rigid body with a fixed point $O$ is described by the following equations
$$m\boldsymbol{\dot v}_S=\boldsymbol F,\quad J_O\boldsymbol{\dot\omega}+\boldsymbol\omega\times(J_O\boldsymbol\omega)=\boldsymbol M_O.$$
Here $\boldsymbol v_S$ is the velocity of the center of mass $S$,
$J_O$ is the tensor of inertia about the point $O$;
$\boldsymbol\omega$ is the angular velocity of the body;
$\boldsymbol M_O$ is a torque applied to the body about the point $O$;
$\boldsymbol F$ is a net force applied to the body.

You know motion of the body so you can find $\boldsymbol F$, $\boldsymbol M_O$. Actually that is all you can find.

If you want to fine the torque then it is convenient to use a moving coordinate frame $Oxyz$ such that $Oz$ is directed upwards and $Ox$ is directed along cone's axis. The vector $\boldsymbol \omega$ is directed along the common generatrix of the cones.
There is also a formula $\boldsymbol v_S=\boldsymbol\omega\times \boldsymbol{OS}$