The Flat Track

• In the diagram, the vehicle with mass m has a velocity v which at any instant in time points in the same direction the front of the car is pointed at that instant. But if the car is traveling in a circle, then there is a centripetal acceleration a_c that points inward along the radius of that circle.
• What force provides this acceleration in this case? Friction between the rubber tire and the roadway.
• For a particular tire compound and roadway surface, there will be a coefficient of static friction (the car is not sliding, yet!) for forces along the radius of the circle.
• The centripetal force ( F_c = ma_c ) then, is supplied by friction and
  1. F_f = ma_c

  Also, so

  2.

From the coefficient of friction formula,

3.

• This reminds us that the frictional force is dependent upon the coefficient of friction. The higher the value of, the greater F_f can become and thus the greater a_c it can supply.
• This in turn implies that there is a maximum value of F_f for a particular value of m, and thus a maximum a_c for a car in this situation.
• Since a_c is also related to velocity and radius by , it follows that there might be a maximum velocity for a car in turn of radius r with a coefficient of friction .

Lets check this out!

4.

• Since the car is neither sinking through the roadway nor accelerating vertically, the normal force and gravitational force (weight) must be in equilibrium,

  5. F_n = mg and we can substitute this value into the previous formula (4) so

  6.

• note that the mass of the car cancels out of the expression such that now
  

  7. and

  8.

• The significance of this is that for any car with the same type of tire on the same road, that is, for any given coefficient of friction, and radius of turn r there is a maximum velocity at which the frictional force will be able to provide the centripetal force needed to keep the car moving in a circle.
• Once this velocity is exceeded, F_f is no longer sufficient to provide the force necessary to produce the resulting centripetal acceleration.
• Static friction along the radius is thus overcome, and since the car can no longer accelerate centripetally, it will move linearly along the velocity vector it was following at the moment static friction was overcome.
• The car is now sliding ( skidding) along the dimension of the radius, and the roadway simply curves out from under the car.
• Of course, this maximum velocity for a particular curve is still dependent upon the coeffient of friction between the tire and the roadway. Is this value constant? NO! It varies because there are all kinds of different compositions of tire material.

So, how do we increase this value of v so that cars can safely go faster around corners and get around the problem of varying values of ? We have to provide some other force than friction in order to provide for the increased a_c.

See banked curves.