49) Obtain an expression for angle of banking of a rough curved road so that a vehicle ca negotiate a turn without sliding off the road. Show that it is independent of the mass of vehicle.
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For the vehicle to go round the curved track at a reasonable speed without skidding, the greater centripetal force is managed for it by raising the outer edge of the track a little above the inner edge. It is called banking of circular tracks.
Consider a vehicle of weight Mg, moving round a curved path of radius r, with a speed v, on a road banked through angleθ.
The vehicle is under the action of the following forces:
The weight Mg acting vertically downwards
The reaction R of the ground to the vehicle, acting along the normal to the banked road OA in the upward direction
The vertical component R cos θ of the normal reaction R will balance the weight of the vehicle and the horizontal component R sin θ will provide the necessary centripetal force to the vehicle. Thus,
R cosθ = Mg …(i)
R sin θ = Mv2r ..........(ii)
On dividing equation (ii) by equation (i), we get
R sin θR cos θ= Mv2/rMgtan θ = v2rg
As the vehicle moves along the circular banked road OA, the force of friction between the road and the tyres of the vehicle, F = μR, acts in the direction AO.
Consider a vehicle of weight Mg, moving round a curved path of radius r, with a speed v, on a road banked through angleθ.
The vehicle is under the action of the following forces:
The weight Mg acting vertically downwards
The reaction R of the ground to the vehicle, acting along the normal to the banked road OA in the upward direction
The vertical component R cos θ of the normal reaction R will balance the weight of the vehicle and the horizontal component R sin θ will provide the necessary centripetal force to the vehicle. Thus,
R cosθ = Mg …(i)
R sin θ = Mv2r ..........(ii)
On dividing equation (ii) by equation (i), we get
R sin θR cos θ= Mv2/rMgtan θ = v2rg
As the vehicle moves along the circular banked road OA, the force of friction between the road and the tyres of the vehicle, F = μR, acts in the direction AO.
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