Question

I tried for an hour but still can't get the answer.

Answer 1

Oh boy,, okay gimme a few

Answer 2

draw a fbd

Answer 3

Draw a free-body diagram when P reaches the top of its trajectory.
It will tell you what its velocity is at this point.

Then you can use conservation of mechanical energy to relate it to U.

Answer 4

The essential piece of physics that gives you the inequality on Usquared, is the fact that the minimum speed at the top of the circle will be such that the tension in the string at that point is zero. In other words, the centripetal force at the highest point is provided by gravity alone.
(If the speed was higher at the top, there would be some tension in the string - if the speed was any lower, the mass would fall below the circular path.)

Then as mentioned in the comment above, you can use energy conservation to relate the condition on the minimum speed at the top, to U.

Answer 5

Clearly the crux of matter is the velocity of the particle at the highest point of trajectory. However, what is equation that relates the centrifugal force to the angular velocity?

Answer 6

I don't think there is any need to introduce the angular velocity, just work with the speed of the particle.
At any point, the centripetal acceleration will be v squared/radius for circular motion.