Centripetal acceleration and forces

Centripetal acceleration describes the motion of object travelling in circles or arcs. It translates to "centre seeking".  An example of this motion is shown below with a student swinging a weight on the end of a string.  Note that if she were to let the string go, it would travel in a straight line with the same speed it has when it was spinning.  This is the instantaneous velocity of the weight.
Now because the motion is circular, while the speed can unchanging, the velocity (a vector quantity) is changing all the time.   Now lets consider this at a time interval started at t1 and finished at t2.

The acceleration of the weight can be calculated by:

When you do the vector subtraction it shows that the force is always directed into the centre of the circle or arc.  The formula for centripetal acceleration is:

It therefore follows that the formula for centripetal force is:

This means that the frictional force of the object on a surface must either exceed or be greater than the centripetal forces or a car will skid off the road when cornering.  This frequently occurs in icy or rainy conditions.

One way this overcome is by banking corners.  Just like hills they unbalance the weight of the car on the road with the reaction force of the road surface on the car.  This in turn means that even if the road was friction-less, the car should be able to take the corner (assuming its travelling below the maximum possible velocity.

The centripetal force contributed by a banked corner can be calculated as:

where theta is the angle of banking from the horizontal.