General motion of a cone on an inclined surface

For a solid cone the COM is $frac{h}{4}$ above its base

Along the incline we can write the following equations.

Forces: $$Mgsintheta+f=Ma$$

Torque: $$frac{fh}{4} = I alpha$$

In this case $I = frac{3}{5}m(frac{r^2}{4}+h^2)$

Rolling: $$alpha = frac{a}{h/4}$$

Since we have 3 equations and 3 unknowns $(f, a, alpha)$ the system can be solved.

"How will the cone move when it starts motion due to its weight"

to answer this equation you have to obtain the equations of motion.

you have 3 generalized coordinates

• rotate about then z -axis angle $psi$
• incline plane x position $s_x$
• incline plane y position $s_y$

thus you get :

Kinetic energy

$$T=frac 1 2,{{it dot{s}_x}}^{2}m+frac 1 2,{{it dot{s}_y}}^{2}m+frac 1 2,{dot{psi}}^{2}{{it J_z} }^{2}-tau_z,psi $$

Where $tau_z$ is the torque due to the friction forces and $J_z$ is the inertia of the cone z component.

Potential energy

$$U=m,g left( sin left( alpha right) {it s_y}+{it s_z} right) $$

Where $s_z$ is the z component of the COM.

Thus: The EOM's:

$$m,ddot{s}_x+F_{Rx}=0$$ $$m,ddot{s}_y+F_{Ry}+m,g,sin(alpha)=0$$ $$J_z^2,ddot{psi}+tau_z=0$$

with $tau_z=F_{Ry},y_squad $ and $F_{Rx}=mu,N,,F_{Ry}=mu,Nquad , N=m,g,cos(alpha)$

According to the EOM's the cone will move "diagonal" on the incline plane and rotate about the cone z- axis