Reconciling the Coriolis Force Contradiction on a Spinning Disk

[Electric to be]

Imagine there is a frictionless disk that spins with angular speed w. There is a ball on it that sits motionless at some radius r from the center. Now, switch to the frame of the rotating disk. In this frame the ball should be spinning with speed w * r. Edit: To be clear, the ball is NOT moving in the original, inertial frame. Only in the non inertial frame.

Since the ball is undergoing apparent circular motion, the net "force" (fictitous included) should be mv^2/r. The centrifugal force provides this force exactly. However, since in this frame the ball also has some velocity relative to the frame, there is also a Coriolis force that is parallel to the centrifugal. As a result the net forces (fictitious included) cannot be mv^2/r. So how is this possible?

At first I thought that the V in the Coriolis force equation must only refer to V in the radial direction. However, wikipedia and other sources said v is simply the total relative velocity. As a result I'm very confused.

Maybe if somebody could show an easy derivation (like on this kind of spinning disk for the Coriolis force) it would help my confusion. However, it still wouldn't reconcile this seeming contradiction.

 

[davenn]

Imagine there is a frictionless disk that spins with angular speed w. There is a ball on it that sits motionless at some radius r from the center. Now, switch to the frame of the rotating disk. In this frame the ball should be spinning with speed w * r.

if the disk is frictionless ... then the ball isn't likely to be put into motion

 

[Electric to be]

if the disk is frictionless ... then the ball isn't likely to be put into motion

That's the point. In the inertial frame, the ball isn't moving. It is only moving upon switching to the non inertial frame of the spinning disk.

 

[davenn]

That's the point. In the inertial frame, the ball isn't moving. It is only moving upon switching to the non inertial frame of the spinning disk.

but with no movement of the ball you can't say it is under centrifugal force or any force for that matter
the spinning disk isn't supplying any force to the ball

really need some one else to explain your reference frames clearer

@DrakkithDave

 

[Electric to be]

but with no movement of the ball you can't say it is under centrifugal force or any force for that matter
the spinning disk isn't supplying any force to the ball

really need some one else to explain your reference frames clearer

@DrakkithDave

Well centrifugal force isn't a real force to begin with, it's fictitious, an artifact from switching to a non inertial frame. In the non inertial frame the ball seems to be accelerating, these fictitious forces are made up to explain this, even though no real forces act on the ball.

 

[willem2]

Imagine there is a frictionless disk that spins with angular speed w. There is a ball on it that sits motionless at some radius r from the center. Now, switch to the frame of the rotating disk. In this frame the ball should be spinning with speed w * r. Edit: To be clear, the ball is NOT moving in the original, inertial frame. Only in the non inertial frame.

Since the ball is undergoing apparent circular motion, the net "force" (fictitous included) should be mv^2/r. The centrifugal force provides this force exactly. .

This is where you go wrong. The centrifugal force is pointed outwards, but the acceleration of the ball is towards the center, so there has to be another force, pointed towards the center, twice the size of the centrifugal force. This happens to be the coriolis force.

 

[Vanadium 50]

I don't understand the confusion. The coriolis force is zero. In the inertial frame, the ball is not moving. In the rotating frame, the ball is traveling in a circle under the influence of the centrifugal force.

 

[willem2]

I don't understand the confusion. The coriolis force is zero. In the inertial frame, the ball is not moving. In the rotating frame, the ball is traveling in a circle under the influence of the centrifugal force.

But the centrifugal force is pointed away from the center. The ball can only move in a circle because of the Coriolis force, which is twice the size of the centrifugal force, and pointed inwards. The coriolis force is -2 Ω x v, where v = Ω r.

 

[Vanadium 50]

I think I don't understand the setup.

 

[willem2]

I think I don't understand the setup.

There's a rotating frame, and a ball that's stationary in an inertial frame, so it's moving in a circle in the rotating frame.
Since there are no external forces on it, the sum of the Centrifugal force and the Coriolis force must account for the centripetal acceleration in the rotaging frame.

 

[Dale]

The setup is as follows: there is a ball which is at rest in an inertial frame, consider it's motion in a rotating frame.

@willem2 has it exactly correct. In the rotating frame the motion of the ball is uniform circular motion, therefore there is a net inwards pointing force. The centrifugal force points outwards. The Coriolis force points inwards with a magnitude exactly sufficient to make the net force the correct direction and magnitude.