ROSTREVOR COLLEGE R O S T R E V O R C O L L E G E S C I E N C E F A C U L T Y G l en S t u ar t R o a d , W o od f o rd e S ou th A u s tr a l ia 5 07 2 T e l e p h on e ( 0 8) 8 30 7 8 20 0 F a c s i m i l e ( 0 8 ) 8 3 0 7 8 3 9 6

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Application 2 Banking of curves

Intended student Outcomes After studying this application you should be able to do the following; Identify the horizontal forces on a vehicle moving with constant velocity on a flat horizontal road. Explain the when a vehicle travels around a banked curve at the correct speed for the banking angle, the horizontal component

of the normal force on the vehicle (and not the frictional force on the tyres) causes the centripetal acceleration. Derive the equation tan = v2/rg relating the banking angle to the vehicle speed v and radius of curvature r. Solve problems involving the use of tan = v2/rg.

CASE 1 Vehicle moving with constant velocity on a flat horizontal road Constant velocity = No net force on vehicle. Weight of vehicle is opposed by the upward

force supplied by the road on the vehicle (total normal force = normal forces on each tyre).

Forward force provided by engine is opposed by drag forces acting toward the rear of the vehicle.

drag forces = frictional forces acting on the non-driving wheels and air drag. CASE 2 Vehicle moving with constant speed in a circular arc on a flat horizontal road. When the vehicle is moving in an arc, there is a

centripetal acceleration and thus force, directed towards the center of the circle.

The force causing the centripetal acceleration is the sideways friction force on the vehicle by the road.

Without this force, vehicles would tend to continue in a straight line rather than turning the corner.

NB. All other forces (mentioned above) remain the same.

F = ma= mv2/r There is a limit to the frictional force provided by

the road and thus a limit to the cornering speed of an arc of given radius.

BMW Hatch

Total normal force on thevehicle by the road

Weight of the vehicle

Total dragforce

Force dueto engine

ROSTREVOR COLLEGE R O S T R E V O R C O L L E G E S C I E N C E F A C U L T Y G l en S t u ar t R o a d , W o od f o rd e S ou th A u s tr a l ia 5 07 2 T e l e p h on e ( 0 8) 8 30 7 8 20 0 F a c s i m i l e ( 0 8 ) 8 3 0 7 8 3 9 6

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CASE 3 Vehicle moving with constant speed in a circular arc on a banked road. Banking of roads is a method for providing the force

necessary to produce a centripetal acceleration for cornering.

Banking of a road can be such that no frictional force is necessary to produce the required centripetal acceleration - allowing for safe cornering in slippery conditions.

On a banked road the normal force is still normal to the road, thus no longer vertical. The vertical and horizontal components of the normal force are shown in the diagram below. The vertical component mg is equal and opposite to the weight of the vehicle. This implies that the normal force is now larger than on a flat road. The horizontal component is directed towards the center of the circle and provides the net force causing the

centripetal acceleration. From the vector diagram:

tan mv2 / r

mgtan v2 / rg

If frictional force is not to contribute to the force

providing the centripetal acceleration, then the angle of banking is determined by the speed of the vehicle and the radius of the curve & is independent of the mass of the vehicle.

Implications for road design (extension) Term used by engineers: Superelevation. Superelevation = tangent of the angle of banking

(symbol, e) e = tan

Restrictions: Roads can only be banked for one speed, despite speed

variations of the many drivers. Highway authorities superelevation constraint, ie. e <

0.1, or = 6o At this rate of superelevation, a road of horizontal

width 10m would be 1m high on the outside of a curve than the inside. The superelevation needed for a car travelling at 100kmhr-1 around a curve of 400m radius, requiring no

frictional force for centripetal acceleration is; e = tan = v2/rg = 0.20 (twice the maximum recommended rate) thus some friction needs to provide the force for the centripetal acceleration.