Analytical treatment of this problem has been made in this report and it has been shown that angle or banking or leaning by the driver in order to avoid skidding of vehicle depends on speed of the vehicle and radius of the bend…
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Analytical treatment of this problem has been made in this report and it has been shown that angle or banking or leaning by the driver in order to avoid skidding of vehicle depends on speed of the vehicle and radius of the bend.Driving around a bend on a road involves either banking of the road or leaning by the driver or a combination of the two; to avoid skidding of the vehicle It was found that this angle in equal to arctan of the ratio of square of the speed of the vehicle and radius of the bend. The detailed treatment is presented in this report. It is a very common experience for the motorcyclist driving around a bend to lean towards the centre of the circle, the bend is arc of which. It is so common that it becomes part of driving instinct. The sharper the bend, more is the inclination and vice versa. Higher is the speed and more is the angle of inclination. Many times, the turn is too sharp and / or the speed is too much to negotiate by the motorists and the vehicle skids causing fatal accidents. A car driver unfortunately cannot lean like a motorcyclist and therefore, he needs something else to help him negotiate a bend without skidding and that is banking of the road. In case of banking, there is an upward slope on the road from inner side towards outer side of the circle. So the question is by what angle a motorcyclist should lean and also, what should be angle of banking on the road near a bend to avoid skidding of a vehicle. These two are essentially same problem and have been treated analytically in the subsequent section. 2. Analytical Treatment A cyclist driving on a straight road is shown in Fig. 1, below. Its weight is balanced by the normal reaction and there is no problem of skidding. Fig. 1: Driving on a Straight Road Fig. 2: Driving along a Bend on a Road  Suddenly a bend comes on the road and he has to move along a circular arc. Moving along a circular arc requires centripetal acceleration and there must be a force to produce this acceleration. If the motorcycle goes on a horizontal circular path, this resultant force will also be horizontal. Let us consider a motorcycle of mass ‘m’ moving at a speed ‘v’ is negotiating a bend of radius ‘r’ and the road is horizontal. Therefore, the external forces acting on the vehicle are the following: (i) Weight of the motorcycle ‘mg’ (ii) Normal reaction force ‘R’ and (iii) Frictional force Ff As the road is horizontal, the normal reaction force ‘R’ is vertically upward. The only horizontal force that can act towards the centre of the circular path is friction Ff. This is static friction and self adjustable. The tyres get a tendency to skid outward and the frictional force which opposes the skidding acts towards the centre. Thus for the safe turn i.e. for the turn without skidding Frictional Force = Centripetal Force or, However, there is a limit to the magnitude of the frictional force Ff. If ?s is the coefficient of static friction between the tyres and the road, the magnitude of the frictional force cannot exceed ?sR. For equilibrium in vertical direction R = mg, therefore, Ff < ?smg Therefore, for a safe turn < ?smg Rearranging we get, ?s > However, one cannot rely on the friction between the road and the tyre to avoid skidding along a bend as this value is limited and varies from location to location and also if the road is wet or dry and on so many different things. Therefore, the motorcyclist tries to lean inward and provide the necessary centripetal force by a component of a normal reaction. Let us assume he leans by an angle ? with horizontal; then the normal reaction will also lean by the same angle (Fig. 3). Now one can see that while the vertical component of normal reaction ‘R’ i.e. RCos? balances the mass of the motorcycle in the vertical direction; the horizontal component RCos? provides the necessary cen
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