## Overturning of Vehicle:

• When a vehicle moves along a curved path with very high speed, then there is a chance of overturning of the vehicle. Inner wheel leaves ground first.

• Let a vehicle of a  mass ‘m’ is negotiating a turn along a curved path of radius ‘r’ with speed ‘v’. Let ‘2a’ be the length of the axle i.e. the distance between the two wheels. Let h be the height of the centre of gravity of the vehicle above the ground. Let R1 and R2 be the reactions on the inner wheel and outer wheel of the vehicle exerted by the road surface.
• The frictional force ‘F’ provides the necessary centripetal force.

• Taking moment of forces about centre of gravity ‘G’

• Reactions are balanced by weight of the vehicle

Solving equations |(1), (2) and (3)

• From these equations, we can see that as the speed increases reaction R1 decreases while reaction R2 increases. When reaction R1 is zero overturning of the vehicle takes place.

This is an expression for maximum velocity of the vehicle by which it can be driven beyond which overturning of the vehicle takes place.

### Notes:

• The angle of banking is independent of the mass of the vehicle. i.e. it is same for the heavy and light vehicles.
• The angle of banking for particular velocity speed of the vehicle is inversely proportional to the radius of the circular road.
• The angle of banking for a particular radius of circular track is directly proportional to the square of the speed of the vehicle.
• The angle of banking changes from place to place, for a particular speed of the vehicle and for a particular radius of circular track, it is inversely proportional to the acceleration due to gravity.
• For particular speed of the vehicle and for a particular radius of circular track, the angle of banking is less on the poles and more on the equator.
• Safe speed for a particular speed of the vehicle angle of banking and for a particular radius of circular track, on the equator, is unsafe on the poles.
• If is the width of the banked road, then the elevation of an outer edge of the road over the inner edge is given by

Provided angle of banking is small. otherwise h = l sinθ

• The necessary centripetal force for negotiating a curve by a vehicle, on the unbanked road is provided by the friction between the road and tyres of the vehicle.
• In actual practice, some frictional forces are always present even on the banked road. So that the actual safe velocity is always greater than the calculated safe velocity on the banked road.