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Banking of a Road Surface to Avoid Skidding
In this context, "banking" has nothing to do with money!  
It refers to the process of tilting a road surface at a certain angle to make it less likely that vehicles will skid or slip across the road when going round a curved part of the road.  
   
Moving in a Straight Line at Constant Speed  
     
No problem.   
   
No tendency to skid off the road.   
   
   
   
Turning on a Horizontal Surface  
     
In order to follow a curved path, there must be a centripetal force to provide the necessary acceleration.  
   
This cannot be provided directly by R or mg because they both act at 90 to the required direction (and can therefore have no components along that direction).  
   
Therefore the required centripetal acceleration must caused by the force of friction, Ff  between the wheels and the road.  
   
If the force of friction is not strong enough, the vehicle will skid (it will "try to" keep going in a straight line in accordance with Newton's first law of motion).  
   
Turning on a Banked Surface  
     
The normal reaction, R, now has a component acting towards the centre of the circular path.  
   
If the angle, θ is just right, the correct centripetal acceleration can be provided by the horizontal component of the normal reaction.  
   
This means that, even if there is very little force of friction, the vehicle can still follow the curved path with no tendency to skid across the road.  
   
   
     
In this diagram, the normal reaction force has been resolved into two perpendicular components, the vertical component, of magnitude given by  
 
and the horizontal component (causing the acceleration), given by  
 
   
   
We will now find an equation to calculate the angle needed for a vehicle to follow the curved path with no tendency to skid.  
Note that this angle will, of course, only be exactly right for a given speed (because the centripetal acceleration required depends on the speed of the body).  
As stated above  
 
but Rh must also be given by  
 
Therefore we have  
  equation 1 
 
The vertical forces are in equilibrium so, considering magnitudes only, we can write  
  equation 2 
 
Dividing equation 1 by equation 2 gives  
 
from which the angle required can be calculated.  
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