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I get that centrifugal force is what appears to force you out of a circular motion if you don't have the centripetal force required to stay in it. For example, a stone being whirled around on a string would go tangentially outward if let go due to its inertia and the fact that nothing's making it go in circular motion anymore.

But in the case of a frictionless object sliding radially outward on a frictionless rotating plate,

(1) Why is it radially outward and not tangential?

(2) Why does it move anyway; the surfaces in contact are frictionless; why doesn't it just stay there as the plate rotates, so that it doesn't move at all? Isn't that what inertia is?

(3) Generally, why does centrifugal force act outward on objects?

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We can analyse the path of the object in a non-rotating inertial reference frame or in a rotating non-inertial reference frame. Note that the physical path in space is the same in each case, but our point of view changes, so our description of the path changes.

As observed in a non-rotating inertial reference frame:

  1. When the centripetal force is present, this causes the object to move in a circle at constant speed. Its velocity vector is always tangential to the circle. When the centripetal force is removed there are no forces acting on the object so it continues to move tangentially in a straight line with constant velocity.
  2. The object continues to move because of inertia.
  3. There is no centrifugal force in an inertial reference frame.

As observed in a rotating non-inertial reference frame:

  1. When the centripetal force is present, this balances the centrifugal force. The net force on the object is zero so it remains at rest. When the centripetal force is removed there is an unbalanced centrifugal force acting on the object. This causes the object to initially move radially outwards. However, since the object is now no longer stationary its path is affected by Coriolis force, so it curves away from a radial path.
  2. The object starts to move because of the unbalanced centrifugal force. Its path is not a straight line because of Coriolis force.
  3. When the motion of the object in a non-rotating reference frame is converted to a rotating reference frame, a radially outward term is introduced because the basis vectors in the rotating reference frame are constantly changing their direction. This term is the centrifugal force.
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  • $\begingroup$ "This causes the object to initially move radially outwards"- why exactly does that need to happen? Shouldn't it be tangential, as dictated by inertia? And can you explain the third point a bit more? $\endgroup$
    – harry
    Commented Feb 15, 2021 at 15:45
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    $\begingroup$ @HarryHolmes If we are describing the path relative to the non-rotating inertial frame then the object moves tangentially to the circle. But if we are describing the path relative to the rotating non-inertial frame then it moves radially outwards (initially) because the centrifugal force (which we need to introduce because we are using a rotating reference frame) is directed radially outwards. For a derivation of the fictitious forces in a rotating reference frame using vector analysis see en.wikipedia.org/wiki/Coriolis_force. $\endgroup$
    – gandalf61
    Commented Feb 15, 2021 at 16:09
  • $\begingroup$ Got it. Just this; why doesn't the object just stay there,stationary with respect to inertial frames, and slip on the rotating plate? Isn't that the outcome which inertia would bring about? $\endgroup$
    – harry
    Commented Feb 15, 2021 at 16:24
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    $\begingroup$ @HarryHolmes In the inertial non-rotating frame the object is not stationary to start with - inertia keeps it moving with constant velocity in a straight line. In the non-inertial rotating frame there is an unbalanced force (centrifugal force) acting on the object so it cannot stay at rest. Same outcome, two different explanations depending on POV. $\endgroup$
    – gandalf61
    Commented Feb 15, 2021 at 16:33
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    $\begingroup$ @HarryHolmes Okay, if the object is at rest initially in the non-rotating frame and there are no forces acting on it then it just stays at rest. $\endgroup$
    – gandalf61
    Commented Feb 15, 2021 at 16:57

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