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How can I determine the magnitude of the vertical (“up”) force that the sphere exerts on the lid, given the radius of the cone, its tangential velocity, and the mass of the sphere (s) (and of course, from that information, the centripetal acceleration experienced by the sphere)?

see figure

I have done the following calculations (shown below) to determine the magnitude of the vertical force, with the help of a professor from school. We made a few assumptions about the rotating sphere in order to simplify the problem, but overall I think the calculations should still approximate a real life situation. I would like to make sure that we have not made any errors, however. Would you guys mind looking at it and making sure our calculations are correct?

Thank you!

∑ FY = (– m*g) – N2M + (N*sinθ) = 0
∑ FX = (N*cosθ) = m*(v2/r)
N2 = (m*g) – (N*sinθ)
(N*cosθ) = (m*v2)/r
N = (m*v2)/(r*cosθ)
N2 = (m*g) – ( ((m*v2)/ (r*cosθ))*sinθ )
N2 = (m*g) – ( ((m*v2)/r) *tanθ)
N2 = m*(g – ((v2/r) *tanθ) )

“Up” Force = N2 = m*(g – ((v2/r)*tanθ) )
Where v equals the tangential velocity of the rotating cone.

Answer 1:

Yup, it looks like everything's almost perfect!

There's one minor detail: when you solved for N2, you switched the sign around. That is, you got
N2 = (m*g) - (N*sinθ) while it should be
N2 = (N*sinθ)- (m*g)
so your final answer is off by an overall minus sign: it should be
“Up” Force = N2 = m*( ((v2/r)*tanθ) - g )

An easy way to check the answer is by consider what are called limiting case:
What happens when certain quantities go to some extreme limits?
For example, say we spin the cone faster, so that v becomes bigger. We expect the ball to get "wedged" in the cone much more tightly, so that the force should become very big and positive. But in your answer, as v gets bigger, the force gets big and negative, which means there was something off about it.

These sorts of things are very handy tricks for checking your answer, and making sure you understood the problem.

Anyway, very nice! I like this problem.



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