Too much radius, not enough circle

A particle of mass 'm' starts performing circular motion on a vertical plane when given a an horizontal velocity U at the lowest point..

This Particle is held by a string of radius r which is fixed from one end. The fixed end is at a height 2r from the ground.

After completing the circle7 times, the particle gets released at an angle \(\theta\) above the horizontal in which the fixed end is present, The particle then performs a projectile motion where the Max height from the ground is 51r/13, and the max height from the point of projection is r.

The particle hits the ground with a* velocity 50m/s, making an angle of 60 degree with the vertical*

Find the minimum length of the string required for this phenomenon to occur. Also find U. And find \(sin \theta\)

Give your answer in the form \(sin\theta\) +|U| + r

Assumptions: Take g=10m/s^2 Neglect air resistance. Round off your 'r' to no decimal places.


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