PLEASE PROVIDE A STEP BY STEP AND EXPLANATION ON HOW TO DO THIS!! I DID IT WRONg

Question

PLEASE PROVIDE A STEP BY STEP AND EXPLANATION ON HOW TO DO THIS!!

I DID IT WRONg AND WANT TO KNOW WHY

if the question is unclear:

A small bead of mass m is constrained to slidw tihout friction inside a circularvertical hoop of radius r which rotates about a vertical axis at an angular speed w.
a.) Draw a force diagram for the bead when it is at an angle theta in equilbrium (what does equilibrium mean for this question?)
b.) express angle theta in terms of m, r, w, and gravitational acceleration g (must first find by getting velocity, but how do you get velocity?)
c.) the magnitude of normal force on the beam due to the hoop

Explanation / Answer

First the forces: mg down, normal force from the wire pushing on the bead toward the center of the hoop (that's it, just two forces)

Now... the bead moves in a horizontal circle with a radius equal to r sin@ (I'm using @ for angle theta), so

horizontal force = centripetal force or

n sin@ = mv^2 / r sin@

Also, vertical forces are balanced, so

n cos@ = mg

Divide the equations

sin@ / cos@ = v^2 / g r sin@

And now... v is the speed of the bead, which can be written as the radius of its circle times w, the angular speed. So

sin@ / cos@ = (w r sin@)^2 / g Rsin@

Simplify...

1 / cos@ = w^2 r / g

Finally

cos@ = g / w^2 r

so theta = inverse cosine (g / w^2 r)

(c) the normal force...

Using result from above   

n cos@ = mg

n (g / w^2 r ) = mg

n = m w^2 r

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