The classic 'stone at the end of a thread problem'

[WorldOfPhysics]

Dear experts,

when a a stone at the end of the string is rotated with a high speed so that the string is suspended making zero degrees with the plane of the ground, how do we analyze the system from the rotating frame.

 

[andrewkirk]

I don't think the string will ever become horizontal. It will asymptotically tend towards horizontal as angular velocity increases.
So the case you are being asked to explain requires no explanation because it doesn't happen.

 

[Redbelly98]

I agree with andrewkirk.

The figure in Case 2 (in post #1) is drawn incorrectly. The correct figure should have the label "T" or tension force for the left-pointing arrow, which should point slightly upward and to the left. Then, the magnitude of the downward force Mg should be much smaller than shown -- equal, in fact to the small upward component of the tension force.

Aternatively, Mg could keep the same magnitude, but the centripetal and tension forces would be drawn much larger than shown in Case 2 -- again with the upward component of tension equal to Mg.

 

[CWatters]

We had a very real world experience of this effect last week when we hired a boat...

The boat had a single main mooring line tied permanently near the front. We entered a lock and moored up by running the mooring line around a vertical pole as shown in the first drawing. The problem came when the lock keeper let water into the lock. This caused a lot of turbulence and it proved impossible for the person holding the free end of the rope to keep us against the bank. The reason was the shallow angle on the rope. The closer the boat gets to the bank the harder it is to pull it any closer.

The answer was to run the rope as shown in the second drawing...

 

[sophiecentaur]

In an ideal case, the tension will be proportional to 1/sin(θ) where θ is the angle between the string and horizontal. As θ approaches 0, the tension will approach Infinity. Your perception of the stone being in the same horizontal plane as the support was, in fact, wrong. i.e. θ was very small and appeared to be zero.
There are many examples where a small sideways force can give a massive increase in tension on a taught rope. This is used very often on boats when a mere human can produce super human force against wind and current. (example mentioned above).