Banking of roads :
To avoid risk
of skidding as well as to reduce the wear and tear of the car tyres
, the road surface at a bend is tilted inwards, i.e., the outer side of road
is raised above its inner side. This is called 'banking of roads'.Consider a car taking a left turn along a road of radius r banked at an angle θ for a designed optimum speed V. Let m be the mass of the car. In general, the forces acting on the car are:(a) Its weight →mg, acting vertically down(b) The normal reaction of the road →N, perpendicular to the road surface(c) The frictional force →fs along the inclined surface of the road.Resolve →N and →fs into two perpendicular components Ncosθ vertically up and →fssinθ vertically down, Nsinθ and →fscosθ horizontally towards the centre of the circular path.If vmax is the maximum safe speed without skidding.
mv2maxr=Nsinθ+fscosθ
=Nsinθ+μsNcosθ
mv2maxr=N(sinθ+μscosθ)....(1)
and
Ncosθ=mg+fssinθ
=mg+μsNsinθ
∴mg=N(cosθ−μssinθ)...(2)
Dividing eq. (1) by eq. (2),
mv2maxr.mg=N(sinθ+μscosθ)N(cosθ−μssinθ)
∴v2maxrg=sinθ+μscosθcosθ−μssinθ=tanθ+μs1−μstanθ
∴vmax=√rg(tanθ+μs)1−μstanθ...,.(3)
This is the expression for the maximum safe speed on a banked road.
At the optimum speed, the friction between the car tyres and the road surface is not called into play. Hence, by setting μs=0 in eq. (3), the optimum speed on a banked circular road is
v=√rgtanθ...(4)
∴tanθ=v2rg or θ=tan−1(v2rg)
From this eq. we see that θ depends upon v,r and g. The angle of banking is independent of the mass of a vehicle negotiating the curve.
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