"v0" means that there are no friction forces at that speed
mgsinΘ = (mv0²/r)cosΘ → the variable m cancels
sinΘ/cosΘ = tanΘ = v0² / gr
Θ = arctan(v0² / gr)
When v > v0, friction points downslope:
mgsinΘ + µ(mgcosΘ + (mv²/r)sinΘ) = (mv²/r)cosΘ → m cancels:
gsinΘ + µ(gcosΘ + (v²/r)sinΘ) = (v²/r)cosΘ
µ = ((v²/r)cosΘ - gsinΘ) / (gcosΘ + (v²/r)sinΘ)
where Θ is defined above.
When v > v0, friction points upslope:
mgsinΘ - µ(mgcosΘ + (mv²/r)sinΘ) = (mv²/r)cosΘ → m cancels:
gsinΘ - µ(gcosΘ + (v²/r)sinΘ) = (v²/r)cosΘ
µ = (gsinΘ - (v²/r)cosΘ) / (gcosΘ + (v²/r)sinΘ)
where Θ is defined above.