Find an expression for the minimum frictional coefficient needed to keep a car with speed v on a banked turn of radius R designed for speed v0.

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23-09-2016
Physics

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Find an expression for the minimum frictional coefficient needed to keep a car with speed v on a banked turn of radius R designed for speed v0.

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taskmasters taskmasters · Answer 1 · 2016-10-07T14:10:12+02:00

"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.