# Centripetal acceleration of a vehicle on banked circular arc

1. Feb 3, 2013

### Zatman

1. The problem statement, all variables and given/known data
A van is moving on a horizontal circular bend in the road of radius 75m. The bend is banked at arctan(1/3) to the horizontal. The maximum speed at which the van can be driven around the bend without slipping is 25m/s. Calculate the coefficient of friction between the road surface and the tyres of the van.

2. Relevant equations
centripetal force = mv2/r
Friction ≤ μR

3. The attempt at a solution
See attached diagram. At the maximum speed F=μR. Resolving perpendicular to the plane gives:

R = mgcosθ
∴ F = μmgcosθ

Resolving in a direction perpendicular to the weight gives:

Rsinθ + Fcosθ = mv2/r
mgsinθcosθ + μmgcos2θ = mv2/r

Solving for μ gives:

μ = [(v2/r)-gsinθcosθ]/[gcos2θ]

And substituting the values gives μ=0.61. This is apparantly incorrect (the answer should be μ=0.40). I cannot see what I have done wrong.

Any hints would be greatly appreciated!

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2. Feb 3, 2013

### haruspex

That would be true if there were no net acceleration perpendicular to the surface. But you know there is a net centripetal acceleration, and that is not parallel to the surface, therefore it has a component perpendicular to the surface.