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General motion of a cone on an inclined surface

Physics Asked by AMM on August 31, 2021

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?

2 Answers

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.

Answered by shrey on August 31, 2021

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"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

Answered by Eli on August 31, 2021

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