Find an expression for the minimum frictional coefficient needed to keep a car with speed v on a banked turn of radius R designe d for speed v0.
1 answer:
"v0" means that there are no friction forces at that speed <span>mgsinΘ = (mv0²/r)cosΘ → the variable m cancels </span> <span>sinΘ/cosΘ = tanΘ = v0² / gr </span><span>Θ = arctan(v0² / gr) </span> <span>When v > v0, friction points downslope: </span> <span>mgsinΘ + µ(mgcosΘ + (mv²/r)sinΘ) = (mv²/r)cosΘ → m cancels: </span> <span>gsinΘ + µ(gcosΘ + (v²/r)sinΘ) = (v²/r)cosΘ </span> <span>µ = ((v²/r)cosΘ - gsinΘ) / (gcosΘ + (v²/r)sinΘ) </span> <span>where Θ is defined above. </span> <span>When v > v0, friction points upslope: </span> <span>mgsinΘ - µ(mgcosΘ + (mv²/r)sinΘ) = (mv²/r)cosΘ → m cancels: </span> <span>gsinΘ - µ(gcosΘ + (v²/r)sinΘ) = (v²/r)cosΘ </span> <span>µ = (gsinΘ - (v²/r)cosΘ) / (gcosΘ + (v²/r)sinΘ) </span> <span>where Θ is defined above. </span>
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