In the proof where we find out minimum velocity such that it completes a rotation, we say that at the topmost point the centripetal force cancels the gravity force and hence tension is zero. Using this we find the horizontal velocity and plug it into the energy conservation equation.
For this case, if the velocity at the topmost point $H$ is $0$ then the bob just falls down. Even if we give the bob some small horizontal velocity at the topmost point it follows a parabolic path and falls down (because the string is not taut). Still, if the velocity at the topmost point is $0$ then velocity at the lowermost point $L$ will be $\sqrt{4gr}$ (by energy conservation principle). That means if we were to give the <PERSON> a horizontal velocity of $\sqrt{4gr}$ at $L$ it would complete its circular motion up to the point some point $P$ between $A$ and $B$, at which tension in the string would be zero, and then fall following a parabolic trajectory. 246
That means there is some minimum horizontal velocity $v_0$ for which the tension of the string will not be zero, bob will not start falling and will continue with its circular motion around the circle. That velocity comes out to be $\sqrt{5gr}$.
However, when it is made to complete one quadrant there is no requirement to calculate a limiting velocity because there is nowhere to follow a parabolic path (there is nowhere to fall even if the tension in the string becomes zero). We can just use the energy conservation equation to calculate the velocity at whatever point we may require. If you don't want to directly apply the equations of energy conservation you could proceed by finding out the equations of motion $$\ddot{\theta}=-\frac{g}{r}\sin \theta$$ This innocent looking differential equation is actually quite nasty and does not have a solution using elementary functions. You could, however, proceed by using $$\ddot{\theta}=\dot{\theta} \frac{d\dot\theta}{d\theta}$$ whose first integral will just give you back the energy conservation equation (or it might take you nowhere!). 499