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 \\\\(\ heta\\\\) 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 \\\\(\\vec {mg}\\\\), acting vertically down(b) The normal reaction of the road \\\\(\\vec {N}\\\\), perpendicular to the road surface(c) The frictional force \\\\(\\vec {f_{s}}\\\\) along the inclined surface of the road.Resolve \\\\(\\vec {N}\\\\) and \\\\(\\vec {f_{s}}\\\\) into two perpendicular components \\\\(N\\cos \ heta\\\\) vertically up and \\\\(\\vec {f_{s}} \\sin \ heta\\\\) vertically down, \\\\(N\\sin \ heta\\\\) and \\\\(\\vec {f_{s}} \\cos \ heta\\\\) horizontally towards the centre of the circular path.If \\\\(v_{max}\\\\) is the maximum safe speed without skidding.\\\\(\\dfrac {mv_{max}^{2}}{r} = N\\sin \ heta + f_{s}\\cos \ heta\\\\) \\\\(= N\\sin \ heta + \\mu_{s} N\\cos \ heta\\\\)\\\\(\\dfrac {mv_{max}^{2}}{r}= N(\\sin \ heta + \\mu_{s} \\cos \ heta) .... (1)\\\\)and \\\\(N\\cos \ heta = mg + f_{s} \\sin \ heta\\\\) \\\\(= mg + \\mu_{s} N\\sin \ heta\\\\) \\\\(\ herefore mg = N(\\cos \ heta - \\mu_{s}\\sin \ heta) ... (2)\\\\)Dividing eq. \\\\((1)\\\\) by eq. \\\\((2)\\\\),\\\\(\\dfrac {mv_{max}^{2}}{r.mg} = \\dfrac {N(\\sin \ heta + \\mu_{s}\\cos \ heta)}{N(\\cos \ heta - \\mu_{s}\\sin \ heta)}\\\\)\\\\(\ herefore \\dfrac {v_{max}^{2}}{rg} = \\dfrac {\\sin \ heta + \\mu_{s}\\cos \ heta}{\\cos \ heta - \\mu_{s}\\sin \ heta} = \\dfrac {\ an \ heta + \\mu_{s}}{1 - \\mu_{s}\ an \ heta}\\\\)\\\\(\ herefore v_{max} = \\sqrt {\\dfrac {rg(\ an \ heta + \\mu_{s})}{1 - \\mu_{s}\ an \ heta}} ...,. (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 \\\\(\\mu_{s} = 0\\\\) in eq. \\\\((3)\\\\), the optimum speed on a banked circular road is\\\\(v = \\sqrt {rg \ an \ heta} ... (4)\\\\)\\\\(\ herefore \ an \ heta = \\dfrac {v^{2}}{rg}\\\\) or \\\\(\ heta = \ an^{-1}\\left (\\dfrac {v^{2}}{rg}\\right )\\\\)From this eq. we see that \\\\(\ heta\\\\) depends upon \\\\(v, r\\\\) and \\\\(g\\\\). The angle of banking is independent of the mass of a vehicle negotiating the curve.