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I have a sphere (mass = 3 kg), constrained to a fixed length rope, rotating (radius = 5 m) on a vertical plane. My textbook ask me about the minimum speed in the highest point in order to keep the circular trajectory.

Now, I know that in the highest point (v=speed):

$$F_c=mg+T$$ $$\frac{mv^2}{r}=mg+T $$ $$ \frac{3}{5}v^2=29,43+T$$ I know that with a low speed I have a low Tension, so let's put T=0 and go on: $$v^2=49,05 \rightarrow v=7$$

The result is correct, but I have a little doubt: if the Tension is 0, why the sphere doesn't go along the tangent or start falling down? Exactly what is forcing the sphere to preserve the circular trajectory?

In my ignorant opinion, the minimum speed is a little more than 7.00 m/s, because the tension mustn't be 0, even a very small number, but not null..

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The tension is only zero for an infinitely short time. Immediately before and immediately after the apex of the swing the tension is non-zero. – John Rennie Nov 5 '12 at 14:09
The tension may be nearly zero at the highest point, but its momentum keeps it going towards its tangent. – prash Nov 5 '12 at 14:32
@JohnRennie but why it follows the tracjectory and doesn't go along the tangent? – Surfer on the fall Nov 5 '12 at 14:36
@prash Exactly, towards its tangent! So it doesn't move along the circular trajectory: it's starting to go away! Am I right? – Surfer on the fall Nov 5 '12 at 14:37
How can it fly away, if it is attached by a rope? It always "wants" to fly tangentially, but the rope stops it from doing so; that's what causes the tension. At the top, if T=0, the rope is not needed for an infinitesimally short time; after that T increases again, and the rope is needed again. If T<0 then yes, the ball will fall down. – hdhondt Nov 6 '12 at 3:12
up vote 2 down vote accepted

Here tension is zero for very sort span of time (infinitesimally sort time), or for an instant only. When ever it moves away from the vertically top position it will fill tension again. So it will move in circular trajectory.
And if we consider instantaneous velocity it is tangential. I think this clarify your doubt.

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Thanks a lot, but the problem is: why will Tension fill again? If tension is 0, we can imagine that there's no bound. So, exactly why the sphere follows the trajectory? – Surfer on the fall Nov 5 '12 at 14:35
Because the sphere is tied to rope. And when the sphere will move out from the vertical top position with tangential velocity of will stretch the string as it will try to move out from the circle. So string will pull it towards the center. That is the tension. – Apurba Nov 5 '12 at 14:42

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