When a vehicle moves along a horizontal unbanked curved road, necessary centripetal force is supplied by the force of friction between the wheels and the surface of the road.
If a vehicle of a mass ‘m’ is moving along a horizontal curved road of radius ‘r’ with speed ‘v’ then,
If μ is the coefficient of friction between the road surface and the wheels then
Necessary centripetal force is supplied by the force of friction.
This expression gives the maximum speed with which a vehicle can be moved safely along a horizontal curved road. If speed is more than this velocity, then there is a danger that the vehicle will get thrown off the road.
Banking of Road:
To make the turning of a vehicle on a curved road safer, the outer edge of the road is raised above the inner edge making some inclination with the horizontal. This is known as banking of road.
Angle of banking: When the road is banked then, the inclination of the surface of the road with the horizontal is known as the angle of banking.
Necessity of Banking:
As the speed of vehicle increases, the centripetal force needed for the circular motion of vehicle also increases. In the case of unbanked road necessary centripetal force is provided by the friction between the tyres and the surface of the road. But there is a maximum limit for frictional force, which depends on the coefficient of friction between the wheels and road.
When the centripetal force needed exceeds the maximum limit of frictional force, the vehicle skids and tries to go off the curved path resulting into an accident.
Without proper friction, a vehicle will not be able to move on the curved road with large speed. To avoid this we may increase the force of friction making the road rough. However, this results in the wear and tear of the tyres of the vehicle. Also, the force of friction is not always reliable because it changes when roads are oily or wet due to rains etc. To eliminate this difficulty, the curved roads are generally banked.
Due to banking of the road, the necessary centripetal force is provided by the component of the normal reaction.
Expression for the Angle of Banking of Road:
Consider a vehicle of mass ‘m’ is moving with speed ‘v’ on a banked road of radius ‘r’ as shown in the diagram. Let ‘θ’ be the angle of banking. The weight “mg” of the vehicle acts vertically downwards through its centre of gravity G, and N is the normal reaction exerted on the vehicle by banked road AC. The angle made by road AC with horizontal AB is called as the angle of banking. Let ‘f’ be the frictional force between the road and the tyres of the vehicle.
The normal reaction N can be resolved into two components (N cosθ) along the vertical (acting vertically upward) and (N sinθ) along the horizontal (towards left) as shown in the diagram. Similarly, the frictional force ‘f’ can be resolved into two components (f sinθ) along the vertical (acting vertically downward) and( f cosθ) along the horizontal (towards left) as shown in the diagram.
Considering equilibrium Total upward force = Total downward force
The horizontal components N sinθ and f cosθ provides necessary centripetal force
This is an expression for the angle of banking for curved road considering friction between the road and tyres of the vehicle.
Case – I:
Case – II:
When the frictional force between the road and tyres of the vehicle is negligible μs = 0.
This is an expression for the angle of banking of a road.
From above equation, it is clear that maximum safe speed of vehicle depends on the radius of the curved road and angle of banking is independent of mass ‘m’ of the vehicle. Thus the angle of banking is the same for heavy and light vehicles. The speed will be maximum when tan θ = 1 i.e. θ = 45°. It means the vehicle can be driven with maximum safe speed only when the angle of banking = 45°.
Factors affecting the angle of banking:
The angle of banking for a curved road is given by
Where, θ = angle of banking, r = radius of the curved road and v = Maximum speed of the vehicle.
The equation clearly indicates that the angle of banking depends on the maximum speed of the vehicle, the radius of the curved road and the acceleration due to gravity g at that place.
The expression does not contain the term ‘m’, which indicates that the angle of banking is independent of the mass of the vehicle. Thus it is the same for both heavy and light vehicles.