# General motion of a cone on an inclined surface

Suppose that a solid cone is placed horizontally on an inclined surface and is initially at rest. How will the cone move when it starts motion due to its weight?

I know that its motion depends on the incline angle and also on the friction coefficient of the surface (as I observed by doing some experiments), but I can't establish the relation between them. Can anyone help me?

• Is friction infinite, finite with stick and slip, or zero? – John Alexiou Jan 21 '14 at 16:11
• The coefficient of friction is not zero, but it can assume any value other than that. – AMM Jan 21 '14 at 16:17
• So the cone rolls sometimes, and slips some other times. This makes the problem much more difficult to deal with. – John Alexiou Jan 21 '14 at 16:40
• The simplest solution is when friction is zero, the cone will slide in a straight line. The 2nd most complex case is pure rolling (infinite friction) and the most complex is slip/roll combination. What where you looking for? – John Alexiou May 29 '15 at 4:28
• Is the cone upright on the surface and sliding, or laying on the surface and rolling. – John Alexiou Jan 10 '20 at 15:54

"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 $$\tau_z=F_{Ry}\,y_s\quad$$ and $$F_{Rx}=\mu\,N\,,F_{Ry}=\mu\,N\quad , N=m\,g\,\cos(\alpha)$$

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

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

Along the incline we can write the following equations.

Forces: $$Mg\sin\theta+f=Ma$$

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.

• Using math formatting by enclosing expressions with $...$ makes any post far more readable. See physics.stackexchange.com/help/notation – John Alexiou May 29 '15 at 4:25
• I'm pretty sure the cone will not move in a straight line, it will try to rotate around its apex while going down the plane. – Shamaz Apr 20 '19 at 20:18