looping the loop 2

You are right but something should be clarified. If v<√(rg) before it reaches the top, the ball cannot maintain a circular motion even before it reaches the top of the track. Hence, the balls will undergo projectile motion before it reach the top of the track. In your figure, it is clear that you think the ball will undergo projectile motion after it reaches the top. I hope this is what your misconception.

Since I have given my answer to this question, I have the duty to clarify the doubt.

There is a condition stated in the original question. The ball could reach the highest point of the track. Following the mathematics given here in this new question, it can be seen that the speed of the ball at the highest point v is at least equal to square-root[rg]. If the ball has speed lower than that value, the ball would not reach the highest point, and which violates the condition given in the original question, a situation that should not be considered.

As such, we only need to consider situations where the ball has speed higher than square-root[rg]. For these situations, the speed of the ball should be higher than square-root[rg] when it rolls downward. Then the equation:

mv^2/r = mg.cos(theta) + R
holds, where "theta" is the angle the radius joining the centre of the track to the ball makes with the vertical.
The equation could be easily complied with for increasing value of R from zero.

Yes, you are right. They may have erroneously assumed that this cause is similar to a roller coaster. For a roller coaster, the car is always stuck to track. Therefore, the car can complete the loop if it has a v > 0 at the highest point.