How Are Friction and Angular Velocity Related in the Cube on Funnel Problem?

[Ahmed Farhan]

I've uploaded a picture instead of writing the whole problem.
So the forces to consider here are - the weight of the cube, the normal force exerted on it by the funnel surface in contact, the centripetal force, and the frictional force.
If there is no relative motion between the cube and the funnel, then the net force must be the centripetal force. But I can't write out the force equations unless I know whether the frictional force points up the incline or down. I'm guessing it has something to do with ω but I can't be sure and I can't find the right logic.
I want to know if my assumptions are right and if they are, how are the friction and ω related ?

 

[ehild]

View attachment 211204
I've uploaded a picture instead of writing the whole problem.
So the forces to consider here are - the weight of the cube, the normal force exerted on it by the funnel surface in contact, the centripetal force, and the frictional force.
If there is no relative motion between the cube and the funnel, then the net force must be the centripetal force. But I can't write out the force equations unless I know whether the frictional force points up the incline or down. I'm guessing it has something to do with ω but I can't be sure and I can't find the right logic.
I want to know if my assumptions are right and if they are, how are the friction and ω related ?

The force of friction can point both up and down the incline. Accordingly, you can get a maximum and a minimum value of ω..

 

[Ahmed Farhan]

The force of friction can point both up and down the incline. Accordingly, you can get a maximum and a minimum value of ω..

Thank you for your answer. But what I was trying to understand is if the frictional force points up the incline, why? And if the opposite is true, I want to know the reason too. I simply don't understand the relation between maximum and minimum value of ω and the direction of friction.

 

[haruspex]

Thank you for your answer. But what I was trying to understand is if the frictional force points up the incline, why? And if the opposite is true, I want to know the reason too. I simply don't understand the relation between maximum and minimum value of ω and the direction of friction.

Other than friction, the only forces acting are gravity and the normal force. To maintain position on the funnel, the resultant force needed is centripetal, so radial. Those three are all in the one plane. Since friction acts to oppose relative motion, there is no cause for the frictional force to act out of that plane. Thus it must be directly up or down the funnel.
If ω were not constant then there would be tangential component to the friction.

 

[ehild]

Thank you for your answer. But what I was trying to understand is if the frictional force points up the incline, why? And if the opposite is true, I want to know the reason too. I simply don't understand the relation between maximum and minimum value of ω and the direction of friction.

Assume the friction is not too high and ω=0, so the funnel is in rest. How does the cube move?
What happens when the funnel rotates very fast?

 

[Ahmed Farhan]

Assume the friction is not too high and ω=0, so the funnel is in rest. How does the cube move?
What happens when the funnel rotates very fast?

So I came to the conclusion after thinking over your suggestion that the friction should point up the incline if ω is small and down the incline if it's large. Is my conclusion correct?

 

[Ahmed Farhan]

Other than friction, the only forces acting are gravity and the normal force. To maintain position on the funnel, the resultant force needed is centripetal, so radial. Those three are all in the one plane. Since friction acts to oppose relative motion, there is no cause for the frictional force to act out of that plane. Thus it must be directly up or down the funnel.
If ω were not constant then there would be tangential component to the friction.

It's becoming clearer to me now but I'm still a bit confused. if ω weren't constant, why would friction have tangential component? And are you implying there's relative motion up or down the incline? Hope this isn't a silly question.

 

[ehild]

So I came to the conclusion after thinking over your suggestion that the friction should point up the incline if ω is small and down the incline if it's large. Is my conclusion correct?

Yes.

 

[haruspex]

if ω weren't constant, why would friction have tangential component?

Because there would be tangential acceleration of the funnel. For the cube to move with it, there would have to be a tangential force on the cube.

 

[vela]

It's becoming clearer to me now but I'm still a bit confused. if ω weren't constant, why would friction have tangential component? And are you implying there's relative motion up or down the incline? Hope this isn't a silly question.

If ω isn't constant, the speed of the block is changing. The radial acceleration, being perpendicular to the motion, can't be the cause; it can only change the direction of the block's movement. There has to be a tangential acceleration parallel to the block's velocity.

 

[Baluncore]

@Ahmed Farhan. Welcome to PF.

Two perpendicular forces act on the mass, m, of the cube, one is radial, due to m and ω, the other vertical due to m and gravity. What is the vector sum of those two forces? What is the direction of that vector? What is the direction of that vector relative to the funnel surface? How does that angle relate to the variable ω ?

Friction acts in the plane of the contact surface. The force on the cube must be resolved into two parts. One component of that force acts normal to the friction surface, it presses the cube onto that surface. The other component acts parallel with the surface. The cube will slide along the friction surface when the parallel component is sufficient to overcome friction.

A coefficient of friction is the ratio of two perpendicular force components.
Do you understand why, and can you explain the importance of; angle = ArcTangent( ratio of forces )?