Driving Around a Bend - Assignment Example

 Summary 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. 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. 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. Contents 1. Introduction 2. Analytical Treatment 3. Numerical Examples 4. Conclusions 5. References 1. Introduction 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[1] Fig. 2: Driving along a Bend on a Road [1] 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 centripetal force for the circular motion along the bend. Therefore, force balance equations along the horizontal and vertical directions are, RSin and RCosq = mg Combining these two equations we get, tan = Thus it can be seen that the required angle of leaning to avoid skidding depends on the speed of the motorcycle and the radius of the bend. For a sharp bend, the radius is small and therefore, for a given speed the motorist will have to lean by a larger angle to avoid skidding. Similarly, for a given radius of a bend, the by which the motorist will have to lean to avoid skidding is more if he speed is more and vice versa. Fig. 3: Force acting on the motorcycle leaned towards the centre by an angle  Banking of a Road near a Bend: While a motorcyclist can afford to lean a car driver cannot and therefore, the road is banked near curves or bends to help a car lean towards the centre. The angle of banking is kept such that for the radius of the bend and the advised speed, the car will not skid. If the speed is slightly more, then frictional forces help to prevent skidding to some extent and beyond certain speed the car skids. 3. Numerical Examples Let us consider a sharp turn on a road of radius equal to 10 m. Assuming that a motorcyclist cannot lean by more than 10o, to what speed he should limit himself so that normal reaction is good enough to prevent skidding or what is the safe speed of driving around such a sharp turn? Let us assume that ‘v’ is the safe speed, then tan = Upon rearranging, v = √(rgtan) = √(10*9.8*tan10o) m/s = 4.2 m/s = 15.12 kmph It can be seen how a turn makes a motorcycle to slowdown for safe driving. 4. Conclusions Based on the analytical treatment of the motion of motorcycle around a bend it can be concluded that frictional force cannot be relied upon to avoid skidding and a motor cyclist must lean towards the centre by an angle that depends on his speed and radius of the bend. 5. References [1] http://www.saburchill.com/physics/chapters/0011.html Read More