Derive or expression for the angle of banking

Ans : 

                               

So first of all , we will discuss about what is angle of banking ? 

The angle of banking is that , the angle made or make by the surface of road with the horizontal surface of road is called as or known as angle of banking .

So let consider a vehicle of mass m is moving with speed v on a banked road , banked at an angle  (θ) as shown in the diagram .

So here let consider F be the frictional force between tyres of the vehicle and the road surface . So here let's learn , the forces acting on the vehicle is ,

i) The weight mg acting vertically downward.

ii) The normal reaction N in the upward direction through C.G .

Here the frictional force in between tyres of the vehicle and the road surface can be resolved by into the, 

F (cosθ)  --  along horizontal direction

F (sinθ)  --  along vertically downward direction .

iii) The normal reaction N can be resolved into two components : 

a) N (cosθ) along vertical direction .

b) N (sinθ) along horizontal direction .

iv) The component N cosθ balances or equalises the weight mg of the vehicle and component F sinθ of the frictional force .

Therefore  N cosθ = mg + F (sinθ)

Therefore  N cosθ  - F sinθ = mg--------(consider equation 1) .

v) The horizontal component N sinθ along with the component F cosθ of frictional forces provides the necessary centripetal force  mv^2 / r .

Therefore   N sinθ + F cosθ = mv^2 / r   ----------(consider equation 2) .

vi) So dividing the equation (2) by (1) ,

So the magnitude of the frictional force depends on the speed of vehicle for a given road surface and tyres of the vehicle .

vii) Let we consider Vmax be the maximum speed of vehicle , and the frictional force produced at this speed given as ,

Now dividing the numerator and denominator of equation (5) by N cosθ , we have ,

viii) For a curved horizontal road θ = 0° , therefore equation (6) becomes ,

ix) Comparing equation (6) and (7) it is concluded that maximum safe speed of vehicle on a banked road is greater than that of curved horizontal road or level road .

x) If μ = 0  , then equation (7) becomes ,

So at this speed , the frictional force is not needed to provided necessary centripetal force . So therefore will be a little wear and tear of tyres , if vehicle is driven at this speed on the banked road , v (not) is called as optimum speed , 

xi) From the equation (8) as we write , 

 xii) Here the equation (9) represents angle of banking of a banked road . Formula for angle of banking does not require mass of vehicle m . Hence angle of banking is independent of mass of the vehicle .

So we hope you understand this derivation of angle of banking with clearly and nicely !