Problem 5) [Textbook 1.27, modified] Objective: Consider the impact of non-inert

Question

Problem 5) [Textbook 1.27, modified] Objective: Consider the impact of non-inertial reference frames on motion (qualitatively In an inertial reference frame, any object subject to zero net force will travel in a straight line at a constant speed. In non-inertial reference frames, an object may appear to travel on a curved path or change speed even though it is subject to zero net force. The following thought experiment will illustrate this. Consider a perfectly flat, horizontal turntable (a flat disk rotating about an axis through its center). The turntable is rotating with constant angular velocity o. You are standing next to the turntable on the ground (in an inertial reference frame). You shove a frictionless puck across the turntable straight towards its center. The puck is subject to zero net force in the horizontal direction and travels in a straight line across the turntable as you observe it Now, please describe the puck's path as observed by someone who is sitting at rest on the turntable. (That observer is rotating along with a fixed point on the turntable which will make the puck appear to deviate from a straight line.)

Explanation / Answer

Let the outside observer be denoted O.

Consider the rotating observer to be at the center of the turntable. Let it be denoted as O'.

His coordinate system is rotating. Let it be an anticlockwise rotation as per O.

Consider a point in the O frame. Let it be at rest.

This point will appear to circle around the origin in O' frame in the clockwise direction.

If we consider polar coordinates, there would be a constant velocity in the theta direction but not in the radial direction. Good.

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Now, let the point move with constant velocity in the O frame.

If O' were not rotating, he would see exactly the same thing as O. But that is simple.

O' is rotating anti clock wise.

Assume the puck is moving from the edge of the turntable to the center. (as seen from the O frame)

If we take polar coordinates in the O' frame, we see that there are two components of velocity:

1) Along the theta direction (as before)

2) A new component along the radial direction, (with reducing radius)

So, if we combine the above two components of the velocity qualitatively, we see that the puck is moving in a circle whose radius is getting smaller and smaller and becomes zero at the center of the turntable.

In other words, the puck is spiralling towards the center.

Now, in the O frame, the puck reaches the center and keeps going on and exits the turntable.

This in the O' frame is simple the exact same situation but with the radius increasing from 0 onwards. The puck is hence seen to spiral away from the origin.

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So, the complete motion is that the puck spirals towards the origin and then spirals away from the origin. This is how O' perceives it.

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