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

[Newton's Protege]

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

 

[OlderDan]

Another thread was already started for this problem with a few added questions.

https://www.physicsforums.com/showthread.php?t=140012

 

[kyle8921]

First and foremost, I want to acknowledge the urgency and importance of this bonus problem for you. I understand the pressure of needing to earn points and the frustration of not knowing how to approach a problem. However, I want to remind you that it is important to stay calm and focused, as that will help you think more clearly and find a solution.

Regarding the problem itself, it is indeed a classic textbook problem that involves the principles of centripetal acceleration and conservation of energy. The key to solving this problem lies in understanding these principles and applying them correctly. I would suggest reviewing the relevant sections in your textbook and seeking help from your instructor or classmates if needed.

In terms of providing a mathematical proof within three lines, that may be a bit challenging. However, I believe it is possible with a clear understanding of the problem and the use of appropriate equations. I encourage you to break down the problem into smaller steps and think about how each step relates to the principles mentioned above.

I have looked at the attached file and it seems to be a helpful illustration of the problem. I suggest using it as a visual aid while solving the problem. Additionally, you can also try searching for similar problems online to get a better understanding of the concept.

In conclusion, I want to remind you to stay calm and focused, review the relevant principles in your textbook, and seek help if needed. With a clear understanding and application of the principles, I am confident that you will be able to solve this bonus problem and earn the much-needed points. Good luck!