[Mechanics] Circular motion outside sphere

AI Thread Summary
The problem involves determining the angle θ at which a marble leaves the surface of a sphere when rolling from the top with zero initial speed. The key point is that the marble will detach from the sphere when the normal force acting on it becomes zero. To solve this, one must apply the principles of conservation of energy and centripetal force at that critical moment. The discussion emphasizes the importance of understanding the relationship between acceleration, velocity, and the forces acting on the marble. The solution requires careful analysis of these dynamics to find the angle of departure.
sushichan
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Homework Statement


A marble rolls from the top of a big sphere. What is the angle θ when the marble is about to leave the sphere? Assume a zero speed at the top.

Homework Equations


The question came from the chapter Mechanics - circular motion.

The Attempt at a Solution


I can't conceive what that instant moment means to the acceleration or velocity of the marble.
 
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I don't really understand your question. Do you have any ideas how to solve this problem?
 
Assumption:A light push is given to the marble.it starts to fall with 0 velocity.
The moment when the ball 'leaves the sphere' is when the normal force between them is 0. Apply conservation of energy and make an expression for centripetal force at the instant when the normal force exterted by the marble is 0.
 

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Kindly see the attached pdf. My attempt to solve it, is in it. I'm wondering if my solution is right. My idea is this: At any point of time, the ball may be assumed to be at an incline which is at an angle of θ(kindly see both the pics in the pdf file). The value of θ will continuously change and so will the value of friction. I'm not able to figure out, why my solution is wrong, if it is wrong .
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