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Asked in NEET 2016 Phase-I · Banked roads and the conical pendulum
Given: banked road of radius R and angle θ, tyre friction μₛ.
Idea: at the maximum speed the car is on the point of sliding up the bank, so friction acts down the slope and adds to the horizontal pull of the normal force.
Along the vertical: N cos θ=mg+μₛN sin θ.
Along the horizontal, towards the centre: N sin θ+μₛN cos θ=(mv²)/R.
Dividing the second by the first removes N and m: (v²)/(Rg)=(sin θ+μₛ cos θ)/(cos θ-μₛ sin θ).
Dividing top and bottom by cos θ: (v²)/(Rg)=(tan θ+μₛ)/(1-μₛ tan θ).
vₘₐₓ=√gR·(μₛ+tan θ)/(1-μₛ tan θ).
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