Question

Formula for maximum velocity on circular path with friction cofficiant

Answer 1

What provides the centripetal force necessary for a car to make a curve on a flat road? What happens if that force is not present?

Gravity pulls down on the car with its weight, w = m g. The level road pushes up with a normal force n and horizontally with a friction force Ffrict.

This is the force of static friction! As the car moves and the tires rotate, the tires are momentarily at rest with respect to the road. Otherwise, the tires skid!

Remember, the friction force can be any value from zero up to a maximum of Fs = s n when the car is just on the verge of sliding. We will consider this case, when the car is just on the verge of sliding. This means

Ffrict = Fs = s n

The diagram at the right shows all these forces. There is no vertical component to the acceleration so we find that

n = m g

which means

Fnet = fs = s m g = m v2/r

s g = v2/r

s = v2/g r

That is, we must have a coefficient of static friction of s = v2/g r to provide the friction force to allow a car, traveling at speed v, to make it around a flat curve of radius r.

Or, we might find the speed in terms of this coefficient of static friction,

v2 = s g r

v = SQRT[s g r]

This is the maximum speed that a curve of radius r can be taken when the coefficient of static friction between tires and pavement is s.

If the velocity increases, the radius r will also increase! This means the car will not follow the intended curve and may run off the road entirely!