What factors affect the motion of a bead on a spinning hoop?

[tiny-tim]

Not sure if this is what you want or not, but subbing Rω²=a_rad into Fx I get mRω²=N*sinθ.

but you're not told what N is!

(and that's not right anyway )

 

[zaper]

Ok so since we don't know what N is and the question doesn't ask for it we need to get rid of it. You said earlier that there is no vertical acceleration so that means N*cosθ=mg or N=mg/cosθ

 

[tiny-tim]

(just got up :zzz: …)

that's right …

now put everything together!​

 

[zaper]

So then plugging N into mrω2=N*sinθ (again r is the horizontal circle's radius) I get:

mrω²=mg*sinθ/cosθ

Simplified this is rω²=g*tanθ

Now I believe that r is R*sinθ so subbing again:

R*sinθ*ω²=g*tanθ which simplifies again to

R*ω²=g/cosθ

Hopefully I did all that right and from here it's easy to get θ by itself

 

[tiny-tim]

R*ω²=g/cosθ

yes!

ok, now you have ω as a function of θ, find both the normal force and the friction (tangential) force as a function of θ, and check whether µ is sufficient

 

[zaper]

Well I already have the equation for N right? (N=mg/cosθ) And can I just use friction≤μN here?

 

[tiny-tim]

Well I already have the equation for N right? (N=mg/cosθ)

no, because you got that by resolving F = ma vertically, in the non-friction case

now, there's a friction force F, which of course will have both horizontal and vertical components, so that equation needs changing

(you may find it more convenient to use R-radial and tangential components instead of horizontal and vertical … I'm not sure, i haven't checked)

 

[zaper]

Ok so with friction for there to be no vertical acceleration then Ny+Fy=mg to stop it from moving down but also mg+Fy=Ny to stop from moving up.

For no horizontal acceleration won't it simply be Nx=Fx?

 

[tiny-tim]

Ok so with friction for there to be no vertical acceleration then Ny+Fy=mg to stop it from moving down but also mg+Fy=Ny to stop from moving up.

nooo, it'll drive you mad if you try doing the two friction cases separately

the only way to stay sane is to just call it F, and find out later whether it's positive or negative

For no horizontal acceleration won't it simply be Nx=Fx?

but there isn't no horizontal acceleration, is there?

 

[zaper]

the only way to stay sane is to just call it F, and find out later whether it's positive or negative

So Ny+F-mg=ma with no vertical acceleration so it's just simply Ny+F=mg

Horizontally then Nx-Fx=ma but where does this acceleration come from?