Bead on a rotating hoop: equilibrium positions

Finding the equilibrium positions of a bead constrained to move on a frictionless rotating hoop. We'll also use physical arguments to decide whether each equilibrium position is stable or not.
About me: I studied Physics at the University of Cambridge, then stayed on to get a PhD in Astronomy. During my PhD, I also spent four years teaching Physics undergraduates at the university. Now, I'm working as a private tutor, teaching Physics & Maths up to A Level standard.
My website: benyelverton.com/
#physics #dynamics #mechanics #mathematics #maths #math #rotation #rotatingframes #centrifugal #force #fictitiousforces #weight #normalreaction #contactforce #equilibrium #frictionless #hoop #bead #constraint #constrainedmotion

[@drandrewsanchez]

love this. I'm personally interested in the classical mechanics and love these examples

[@DrBenYelverton]

Excellent, I'll try to keep the examples coming!

[@instantchanel9696]

Thank you so mach 😁 it really help my preparation for national olympiad

[@DrBenYelverton]

Good to hear, thanks for watching!

[@nikosathanasopoulos5257]

Found that the angular frequency of small oscillations around the stable equilibrium points, when ω>sqrt(g/a) is ω0^2=(ω^4-ωc^4)/ω^2, where ωc^2=g/a. Not so sure but I'm open to any other other solutions.

[@canopus4]

Rotating hoop problem from David Morin's book

[@danielpe8533]

hi, thanks that was realy helpfull! I would just like to know, because that's something I didn't realy understand- why you shouldn't include N in the equation of newton's second law?

[@DrBenYelverton]

I'm glad it was helpful! The force N doesn't appear in the equation because we resolved in the tangential direction. It's a purely radial force, so doesn't have any component in the tangential direction.