In the question to find out the minimum speed of a stone fastened to a string at the top most point in a vertical circle, we consider tension $T$ to be zero and equate $mv^2/r$ to $mg$. But isn't it the tension which provides the centripetal force $(mv^2/r)$? When we consider this to be zero, how come do we equate centripetal force to gravitational force?
Also, the equation $T-mg=mv^2/r$ holds for the top most point. How is this true when we consider $mv^2/r$ to be centripetal force whose direction is always towards the center? If this equation holds true, won't this mean that tension is 'pushing' the stone outwards when tension is considered a pulling force?
Best Answer
First, some information to motivate my answer.
In circular motion, it is useful to break forces into components: the radial component $F_r$ (points towards or away from the center of the circle around which the particle is moving) and the tangential component $F_\theta$ (points tangent to the circle around which the particle is moving).
For planar motion, we can derive equations of motion for these components: $$\mathbf a=(\ddot r-r\dot\theta^2)\hat r+(r\ddot\theta+2\dot r\dot\theta)\hat\theta$$ where $(r,\theta)$ is the polar coordinate, and a dot above a variable represents a time rate of change (so, for example, $\dot r$ is just the time rate of change of $r$).
The case you are interested is circular motion, where $\dot r$ and $\ddot r$ are both $0$. Then we have $$\mathbf a=-r\dot\theta^2\hat r+r\ddot\theta\hat\theta$$
Therefore, using Newton's second law and using the relation $\dot\theta=v/r$, we have $$F_r=-\frac{mv^2}{r}$$ where the negative sign represents the fact that the net radial force component needs to point towards the center of the circle.
I want to point out to you a key thing in this line of reasoning. Instead of thinking of "which force is 'the centripetal force'?", instead all we need to do is determine all of the radial force components, and then we just set this net radial component equal to $mv^2/r$ towards the center of the circle. There is no single "centripetal force" in general.
So now, let's look at your question:
If we are looking at this stone moving in a vertical circle, and if we want to analyze the forces acting on the stone at the top of this circle, we need to consider the radial force components. Well, we know that the entire tension force acts towards the center of the circle, and we know that the weight of the particle acts downwards, which here is also towards the center of the circle. Therefore, based on the argument above it must be that $$-T-mg=-\frac{mv^2}{r}$$
Now, what if we want to think about the minimum speed the stone needs to have at the top of the circle to maintain circular motion? Well, the weight $mg$ is constant for our stone, so it must be that the speed $v$ determines the tension $T$ (or you could argue it is the other way around). Based on our above equation, we have $$v^2=\frac{(T+mg)r}{m}$$ You should convince yourself that based on this equation we can say that the minimum value of $v$ must mean that $T$ is also at its minimum value. Since the tension force cannot become a force that points away from the circle, it must be that the minimum of $T$ is $0$. Therefore, to determine the minimum velocity needed at the top of the circle for circular motion to still occur, it must be that $T=0$.
As for your second paragraph, you should now be able to see that the equation you have written is incorrect. However, that equation would be correct for looking at the bottom of the circle. But in this case tension cannot be $0$ (can you see why based on the information in this answer?).
Based on everything above, I want to specifically address your questions now
Tension does always point towards the center of the circle, but this does not mean it is the only force that has a radial component. The weight of the stone also has a radial component (except for at points where the rope is horizontal and the weight is completely tangent to the circle). In general you should start with $F_r=-mv^2/r$ and then determine what $F_r$ is.
Based on the above discussion, it should be obvious that since $T+mg=mv^2/r$ at the top of the circle, it must be true that $T=0$ means $mg=mv^2/r$. This makes sense, because now the only force acting on the stone is gravity, and this force only has a radial component.
I have corrected the statement to correctly say this holds at the bottom of the circle. A better way to view this equation is $T=mg+mv^2/r$. This shows that at the bottom of the circle the tension force (still pointing in towards the center of the circle) is actually greater than just the weight of the stone.