Questions about a particle shot off of a rotatiting disc.

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A rotating disc with angular velocity w has a particle dropped at half the radius, experiencing centripetal acceleration of Ac(@.5r) = rw^2/2. To determine how long it takes for the particle to be shot off the disc, one must consider that the particle, when dropped perpendicularly, does not move due to the assumption of no friction. The acceleration for circular motion at radius (r/2) is a = 2rw^2, but without initial tangential velocity, the particle remains stationary. Consequently, the particle will not exhibit any motion, including circular motion, while on the disc. The scenario illustrates the importance of initial conditions in analyzing motion on a rotating system.
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Say you have a disc that rotates with angular velocity w. Assume that you know the value of w and the radius of the disc. You drop a particle on the disc while it is rotating half way out from the center of the radius. So the the centripetal acceleration on the particle is

Ac(@.5r) = rw^2/2

How can you go about finding how long in terms of rotations will it take the particle to be shot off the disc ignoring friction. This is not a specific problem just somthing I have been thinking about.
 
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first of all, the accerlation the ball feels if it were to move in circular motion with radius (r/2) would be a = 2rw^2

second, if you drop ball directly onto the disc, ie. the trajectory of the fall is perpendicular to the plane of the disc, then the ball would not move, because you told us to ignore friction.

In other words, the ball wouldn't exhibit circular motion, let alone motion at all.
 
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