What Is the Angle Where a Mass Leaves a Frictionless Sphere?

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The problem involves determining the angle at which a mass leaves a frictionless sphere after sliding down. The solution requires understanding centripetal acceleration and conservation of energy principles, leading to the conclusion that the angle is 48 degrees. A mathematical proof can be succinctly presented in three lines. This problem is recognized as a classic textbook scenario, though it is not included in the user's textbook. Additional resources and discussions are available in related forum threads for further assistance.
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My instructor has assigned me a bonus problem that I have no idea how to do. It is worth 25 points and I need these points very badly! I would be so grateful if anyone knows how to do this problem.

A mass sits on a frictionless sphere. The mass slides down the sphere to a certain point. The mass then acts like a projectile at that point. What is the angle of the point where the mass leaves the sphere. This angle is 48 degrees. Centripetalal acceleration is involved and so is conservation of energy. The instructor wants a mathematical proof of this. He said this can be accomplished within three lines on a sheet of paper. Apparenly, it is a classic "textbook problem" but it is absent from my textbook. I have enclosed an attached file to better illustrate what is happening
 

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