Understanding centrifugal force

[WorldOfPhysics]

Dear Experts,
I believe that we use the concept of Pseudo Forces, to analyze mechanics within an accelerating frame of reference. Pseudo force seems to be a 'correction' in acceleration provided to all the points that are not riveted to the accelerating frame of reference. And centrifugal force is the pseudo force when we consider a rotating frame of reference.

Assuming a truck with a friction-less carrier lodging a block, when the truck accelerates forward, it is easy to understand that the block which has no reason to move along with the truck stays where it is, and to an observer in the frame of reference of a truck feels that the block is moving away from the truck with an equal acceleration.

At the same time, considering a spaceship , spinning at a particular angular velocity, and all the occupants aloft inside it, it is hard to comprehend how they get pushed to the edges/walls of the spaceship.What propels them to the edges? Or how can we understand the object being driven to the edges?

 

[Dale]

Hi WorldOfPhysics, welcome to PF!

Are you asking for a description in terms of an inertial frame or in terms of the rotating frame?

 

[WorldOfPhysics]

It would be highly helpful if i could understand it from both frames.
Thank You.

 

[Dale]

OK, first from the inertial frame.

In that frame there is nothing pushing any of the occupants towards the edge (neglecting air resistance). If an "aloft" occupant, call him "Rob", is at rest in the inertial frame then Rob will remain at rest as the spaceship spins around them. If an "aloft" occupant, call him "Mike", is moving in the inertial frame then Mike will move in a straight line at a constant velocity until Mike collides with the wall of the ship. At that point then the wall of the ship will exert a force on Mike and accelerate him, typically both tangentially and centripetally, never centrifugally.

Does that make sense? Any questions about the description in the inertial frame?

 

[WorldOfPhysics]

Thank you Sir, and i really appreciate your guidance,

So, i believe that from an inertial frame, an observer will find no effect what so ever on an aloft body, unless it strikes on the wall. When it strikes on the inner walls, it receives a normal reaction 'radially inward' which i think is the centripetal force that is imparted to the body. Could you please elaborate on the tangential force/acceleration that gets imparted on the body.

 

[Dale]

Could you please elaborate on the tangential force/acceleration that gets imparted on the body.

Certainly. The force from the wall is not just a normal force, there is a normal force and also a frictional force. The normal force points radially inwards, and is therefore called "centripetal". The friction force from the wall points parallel to the wall and is therefore aligned along a tangent line to the floor. It is therefore called "tangential". The words "tangential" and "centripetal" describe the direction of the force, but the actual forces are due to the friction with the wall and the normal force from the wall respectively.

 

[WorldOfPhysics]

So is it correct that , one must exclude the tangential force, if we assume that the wall is friction-less. In which case, only the normal reaction is felt by the body.

in both cases, how will the body be affected. the normal reaction will possibly give the body the centripetal force required to keep in in the circular track, but what difference could the presence or absence of friction make in this case?

 

[Dale]

So is it correct that , one must exclude the tangential force, if we assume that the wall is friction-less. In which case, only the normal reaction is felt by the body.

Yes.

in both cases, how will the body be affected. the normal reaction will possibly give the body the centripetal force required to keep in in the circular track, but what difference could the presence or absence of friction make in this case?

Yes, the normal reaction can give the centripetal force required for uniform circular motion, however in the absence of friction Mike's angular speed will generally be different than the angular speed of the spaceship (i.e. Mike will slide around the inner wall somewhat like a ball sliding around a roulette wheel). For Mike's angular momentum about the axis of the ship to change there must be a torque on Mike, and the normal force does not provide any torque since it is directed towards the axis.

 

[WorldOfPhysics]

I understand the fact that in the absence of frictional force to provide a tangential force, there is no torque on the body about the center of the spinning ship. In that case, could we still find out a clear idea about how 'mike' will move as observed from an inertial frame. Purely under the influence of normal reaction only.

Considering the small piece of the wall of the spaceship on to which mike strikes , i think it is fair to assume that the binding and strength of the material gives it the required centripetal force to remain in the circular path in sync with the rest of the space ship.

Is it possible to compare that force with the normal reaction mike feels from the wall..I mean compare in some degree of clarity

 

[Dale]

Yes. Mike will initially move in a straight line at a constant speed. He will thus have both a constant linear momentum and a constant angular momentum until he collides with a wall. At that point the normal force will constrain him to move in uniform circular motion at the same constant angular momentum as previously.

 

[WorldOfPhysics]

Sir, i think i get u on this. Please tell me if i am wrong.

If there is no frictional interaction, Its just like a collision. During that collision, Mike gets some momentum imparted to him. and he should move in a straight line in the direction of the final momentum(his initial plus what the normal reaction imparted on him during the short time on collision)

 

[Dale]

That is an interesting approach. Yes, I think that would work. I was implicitly assuming a plastic collision, but in principle he could bounce elastically off the wall instead.

 

[WorldOfPhysics]

Still, from an inertial observer, assuming a friction less Ship wall vs Mike interaction, what i understand is that there is no element of rotational effect on him. Mike never seem to execute a circular motion, in any case. or even in sync or out of sync with the ship wall. And it doesn't seem as though he is stuck or attached or pushed to the wall from an observer in an inertial frame outside the ship.

 

[Dale]

Mike never seem to execute a circular motion, in any case. or even in sync or out of sync with the ship wall.

Assuming a plastic collision then Mike would travel in uniform circular motion after hitting the wall. He would just slide along the wall at a constant angular rate, typically out of synch with the ship wall.

Have you seen a ball roll around a roulette wheel?

 

[WorldOfPhysics]

Is a plastic collision possible if there is no frictional interaction to reduce the kinetic energy?

 

[Dale]

Is a plastic collision possible if there is no frictional interaction to reduce the kinetic energy?

Yes. The normal force can slow Mike even without friction. Think about a car colliding head on with a wall.

 

[Dale]

Are you comfortable with the inertial frame? If so then we can look at the rotating frame. However, I would not want to analyze the frictionless scenario from the rotating frame, only a Rob scenario or a Mike who sticks to the wall scenario.

 

[WorldOfPhysics]

how would the same case be analyzed from the rotating frame of reference

 

[Dale]

In the rotating reference frame it is important to remember that there are two fictitious forces at work both of which are proportional to the rotation rate.

The first is the centrifugal force. It points away from the axis of rotation and is proportional to the distance from the axis. It affects all objects.

The second is the Coriolis force. It affects only objects which are moving in the rotating frame. It is perpendicular to both the axis of rotation and the velocity and it is proportional to the velocity.

So, Rob is at rest in the inertial frame but is moving in the rotating frame. Therefore he is subject to the centrifugal force and also the Coriolis force. The Coriolis force is centripetal and is greater than the centrifugal force. The net force of those two therefore causes Rob to orbit around the axis.

On the other hand Mike is not moving at the right velocity for the Coriolis force to be purely centripetal. Therefore, he is not orbiting the axis but undergoing some other motion. This will cause him to have a curved trajectory which eventually hits the wall. Once he hits the wall the friction will slow him down until he is at rest. At that point there will be no more Coriolis force and the normal force will exactly equal the centrifugal force.