Particle sliding down a sphere - When does it leave the sphere?

[Adoniram]

Homework Statement

A particle is placed on top of a smooth (frictionless) sphere of radius R. If the particle is slightly
disturbed, at what point will it leave the sphere?

Homework Equations

Same as first question, just
F = ma = ΣF_i

The Attempt at a Solution

Similarly, we want to know when the normal force of the sphere on the particle is overcome:
F_norm = F_cent
mg CosΘ = (mv2/r)
CosΘ = y/R (where y is the height above the center of the sphere)

So:
y = v2/g

Finding v2:
Using conservation of energy, PE_initial = PE_final + KE_final
mgR = mgy + mv2/2

Solving for v2
v2 = 2g(R-y)

Placing into equation for y:
y = 2g(R-y)/g = 2(R-y)

Solving for y:
y = (2/3) R

Correct? Or am I making a horrible mistake?

 

[PhotonSSBM]

This is correct.

 

[ehild]

The title of the thread is "Ball rolling down a sphere". You solved the problem of a particle sliding down a sphere instead of a ball rolling down.

 

[Adoniram]

You're right, if I could change the original post, I would, but there's no "Edit" button. Could be an account permissions issue...

 

[berkeman]

I can fix it. So the title should read "particle" instead of "ball"?

 

[Adoniram]

Yes, thank you, and an even better description would be "Particle sliding down a sphere" (instead of rolling)

 

[berkeman]

Done! And thank you to @ehild -- I had the same question when I saw the thread and posted answer.