(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I'm stuck on a relatively simple problem. I can't use Lagrangian mechanics, only Newtonian one.

A mass m slides without friction over a sphere of radius r and mass M. Find its period and the allowed values of its speed.

2. Relevant equations

The sphere isn't in an external gravitational field. So there's the gravitational force between the mass m and the sphere.

3. The attempt at a solution

I found out the period of the mass to be [itex]T=\frac{2\pi R}{v}[/itex] where v is the speed of the mass.

And now this is where I'm stuck. I know that the modulus of the gravitational force between the mass and the sphere is [itex]F_g=\frac{GMm}{R^2}[/itex]. I also know that for a critical value of v, the mass will start to be in orbit over the sphere. This happens when the normal force is worth 0N. So I think I must express the modulus of the normal force acting on the mass in function of the speed of the mass. But I don't know how to "include v" in the expression for the normal force. Hmm.

Is that a reasonable way to approach the problem? Could you give me any tip? Thanks in advance.

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# Mechanics problem, mass sliding over a sphere

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