Banking Of Roads 1x1560

Banking of roads Consider a vehicle of mass 'm' is moving along a horizontal curve road of maximum speed 'v'. The radius of curvature of road is 'r'. When the vehicle is moving along the horizontal curve road, the necessary centripetal force is provided with the frictional force between surface of road and tyres of vehicle. In absence of this force, a vehicle is tangential thrown off from the road. Centripetal force = Force of friction

Where μ is a coefficient of force of friction.

Thus from the equation , it is concluded that maximum safety speed of a vehicle along the horizontal curve road is independent of mass of vehicle, but it depends upon. 1. Coefficient of force of friction. 2. Radius of curvature of road. 3. Acceleration due to gravity. Necessary of banking of roads The

safety

speed

of

the

vehicle

along

the

horizontal

curved

road

is,

1. The safety speed of the vehicle depends upon the coefficient of force of friction. The force of friction between the tyres and road surface is not reliable, it changes due to the nature and condition of the road i.e. rough and smooth. It also changes in rainy season. So the force of friction will not provide the necessary centripetal force for the safety turning. Hence the vehicle will skid or over turn the road. 2. The maximum force of friction also provides wear and tear of the tyres due to making a road surface rough. To ensure the safety speed of the vehicle along the h o r iz o n t a l b a n k e d c u r v e d r o a d t h e ho r iz o n t a l component of normal reaction acting on the vehicle provides the necessary centripetal force for safe turning. Thus the vehicle moves safely along the banked curved road.

Banking of roads : The arrangement of road in which outer edge of the horizontal curve road is raised over the inner edge of the road is called banking of roads. Angle of banking : The angle of inclination of the road surface with the horizontal surface of road is called angle of banking. Expression for angle of banking Consider a vehicle of mass 'm' is moving along a horizontal banked curve road of radius of curvature 'r'. The maximum safety speed of a vehicle along the banked curved road is 'v'. Let 0 be the angle of banking of the curved road.

The frictional force between tyres of vehicle and road surface can be resolved into, i) fcos θ - along horizontal direction, ii) fsinθ - along vertically downward direction. The normal reaction force is greater than weight of the vehicle. The normal reaction force is resolved into two components. The component N cos θ of normal reaction is balanced by weight mg of vehicle and component f sin 0 of frictional force, N cos θ = mg + f sin θ mg = N cos θ - f sin θ H o r i z o n t a l c o m p o n e n t N s i n θ a l o n g w i t h component f cos θ of frictional force provides the necessary centripetal force.

N sin θ + f cos θ =

-------------------(1)

=N sin θ + f cos θ

---------------------(2)

The magnitude of frictional force depend on speed of vehicle for given road surface and tyres of vehicle. Let v ma x denote maximum speed of vehicle the frictional force produced at this speed should be,

----------------------------------(3) For a horizontal curved road,θ = 0°, hence equation becomes,

Combining equation (1)&(2)maximum safe speed of vehicle on a banked road is greater than that of horizontal curved road. If us= 0, then equation becomes,

---------------------------------(4) At this speed the frictional force is not needed to provide necessary centripetal force. There will be a little wear and tear of tyres, if vehicle is driven at this speed on banked road. Vo is called as optimum speed. From equation (4) we can write, Thus angle of banking is independent of mass of vehicle. This is the required expression for safety speed of a vehicle along the banked curved road. From the above equation it is concluded that safety speed of a vehicle is independent of mass of a vehicle but it depends on 1. radius of curvature of road.

2. acceleration due to gravity at that place. 3. angle of banking of road. Thus safety speed of vehicle is same for any vehicle along the banked curved road.

Height of banking : Ellevation Raising of outer edge above the inner edge is called elevation. Let lbe the width of the road and h be the height of banking.

I n o r d e r t o a v o i d a c c i d e n t s t h e f o l l o w i n g precautions should be taken when a motor vehicle moves over a curved path or track. 1.The curved path should be banked (tanθ = v2/rg) 2.The distance between the rails or wheels should be large. 3.The height of centre of mass of the vehicle from the earth surface should be small. 4.The velocity of the vehicle should be decreased at the curved path.