One often hears of "centrifugal force." This is the apparent force that throws you to the outside of a turn during cornering. If
there is anything loose in the car, it will immediately slide to the right in a left hand turn, and vice versa. Perhaps you have
experienced what happened to me once. I had omitted to remove an empty Pepsi can hidden under the passenger seat. During a
particularly aggressive run (something for which I am not unknown), this can came loose, fluttered around the cockpit for a while,
and eventually flew out the passenger window in the middle of a hard left hand corner.
I shall attempt to convince you, in this month's article, that centrifugal force is a fiction, and a consequence of the fact first
noticed just over three hundred years ago by Newton that objects tend to continue moving in a straight line unless acted on by an
When you turn the steering wheel, you are trying to get the front tyres to push a little sideways on the ground, which then pushes
back, by Newton's third law. When the ground pushes back, it causes a little sideways acceleration. This sideways acceleration is a
change in the sideways velocity. The acceleration is proportional to the sideways force, and inversely proportional to the mass of
the car, by Newton's second law. The sideways acceleration thus causes the car to veer a little sideways, which is what you wanted
when you turned the wheel. If you keep the steering and throttle at constant positions, you will continue to go mostly forwards and
a little sideways until you end up where you started. In other words, you will go in a circle. When driving through a sweeper, you
are going part way around a circle. If you take skid pad lessons (highly recommended), you will go around in circles all day.
If you turn the steering wheel a little more, you will go in a tighter circle, and the sideways force needed to keep you going is
greater. If you go around the same circle but faster, the necessary force is greater. If you try to go around too fast, the adhesive
limit of the tyres will be exceeded, they will slide, and you will not stick to the circular path-you will not "make it."
From the discussion above, we can see that in order to turn right, for example, a force, pointing to the right, must act on the car
that veers it away from the straight line it naturally tries to follow. If the force stays constant, the car will go in a circle.
From the point of view of the car, the force always points to the right. From a point of view outside the car, at rest with respect
to the ground, however, the force points toward the centre of the circle. From this point of view, although the force is constant
in magnitude, it changes direction, going around and around as the car turns, always pointing at the geometrical centre of the
circle. This force is called centripetal, from the Greek for "centre seeking." The point of view on the ground is privileged, since
objects at rest from this point of view feel no net forces. Physicists call this special point of view an inertial frame of
reference. The forces measured in an inertial frame are, in a sense, more correct than those measured by a physicist riding in the
car. Forces measured inside the car are biased by the centripetal force.
Inside the car, all objects, such as the driver, feel the natural inertial tendency to continue moving in a straight line. The driver
receives a centripetal force from the car through the seat and the belts. If you don't have good restraints, you may find yourself
pushing with your knee against the door and tugging on the controls in order to get the centripetal force you need to go in a circle
with the car. It took me a long time to overcome the habit of tugging on the car in order to stay put in it. I used to come home with
bruises on my left knee from pushing hard against the door during an autocross. I found that a tight five-point harness helped me to
overcome this unnecessary habit. With it, I no longer think about body position while driving - I can concentrate on trying to be
smooth and fast. As a result, I use the wheel and the gearshift lever for steering and shifting rather than for helping me stay put
in the car!
The 'forces' that the driver and other objects inside the car feel are actually centripetal. The term centrifugal, or "centre
fleeing," refers to the inertial tendency to resist the centripetal force and to continue going straight. If the centripetal force
is constant in magnitude, the centrifugal tendency will be constant. There is no such thing as centrifugal force (although it is a
convenient fiction for the purpose of some calculations).
Let's figure out exactly how much sideways acceleration is needed to keep a car going at speed v in a circle of radius r. We can
then convert this into force using Newton's second law, and then figure out how fast we can go in a circle before exceeding the
adhesive limit-in other words, we can derive maximum cornering speed. For the following discussion, it will be helpful for you to
draw little back-of-the-envelope pictures (I'm leaving them out, giving our editor a rest from transcribing my graphics into the
Consider a very short interval of time, far less than a second. Call it dt (d stands for "delta," a Greek letter mathematicians
use as shorthand for "tiny increment"). In time dt, let us say we go forward a distance dx and sideways a distance ds. The forward
component of the velocity of the car is approximately v = dx / dt. At the beginning of the time interval dt, the car has no sideways
velocity. At the end, it has sideways velocity ds / dt. In the time dt, the car has thus had a change in sideways velocity of ds / dt.
Acceleration is, precisely, the change in velocity over a certain time, divided by the time; just as velocity is the change in
position over a certain time, divided by the time. Thus, the sideways acceleration is
How is ds related to r, the radius of the circle? If we go forward by a fraction f of the radius of the circle, we must go sideways
by exactly the same fraction of dx to stay on the circle. This means that ds = f dx. The fraction f is, however, nothing but dx / r.
By this reasoning, we get the relation
We can substitute this expression for ds into the expression for a, and remembering that v = dx / dt, we get the final result
This equation simply says quantitatively what we wrote before: that the acceleration (and the force) needed to keep to a circular
line increases with the velocity and increases as the radius gets smaller.
What was not appreciated before we went through this derivation is that the necessary acceleration increases as the square of the
velocity. This means that the centripetal force your tyres must give you for you to make it through a sweeper is very sensitive to
your speed. If you go just a little bit too fast, you might as well go much too fast - you're not going to make it. The following
table shows the maximum speed that can be achieved in turns of various radii for various sideways accelerations. This table shows
the value of the expression
which is the solution of a = v2 / r for v, the velocity. The conversion factor 15/22 converts v from feet per second to miles per
hour, and 32.1 converts a from gees to feet per second squared. We covered these conversion factors in part 3 of this series.
Table 1. Speed (Miles Per Hour)
For autocrossing, the columns for 50 and 100 feet and the row for 1.00G are most germane. The table tells us that to achieve 1.00G
sideways acceleration in a corner of 50 foot radius (this kind of corner is all too common in autocross), a driver must not go faster
than 27.32 miles per hour. To go 30 mph, 1.25G is required, which is probably not within the capability of an autocross tyre at this
speed. There is not much subjective difference between 27 and 30 mph, but the objective difference is usually between making a
controlled run and spinning badly.
The absolute fastest way to go through a corner is to be just over the limit near the exit, in a controlled slide. To do this,
however, you must be pointed in just such a way that when the car breaks loose and slides to the exit of the corner it will be
pointed straight down the optimal racing line at the exit when it "hooks up" again. You can smoothly add throttle during this
manoeuvre and be really moving out of the corner. But you must do it smoothly. It takes a long time to learn this, and probably a
lifetime to perfect it, but it feels absolutely triumphal when done right. I have not figured out how to drive through a sweeper,
except for the exit, at anything greater than the limiting velocity because sweepers are just too long to slide around. If anyone
, perhaps?) knows how, please tell me!
The chain of reasoning we have just gone through was first discovered by Newton and Leibniz, working independently. It is, in fact,
a derivation in differential calculus, the mathematics of very small quantities. Newton keeps popping up. He was perhaps the greatest
of all physicists, having discovered the laws of motion, the law of gravity, and calculus, among other things such as the fact that
white light is made up of multiple colours mixed together.
It is an excellent diagnostic exercise to drive a car around a circle marked with cones or chalk and gently to increase the speed
until the car slides. If the front breaks away first, your car has natural understeer, and if the rear slides first, it has natural
oversteer. You can use this information for chassis tuning. Of course, this is only to be done in safe circumstances, on a rented
skid pad or your own private parking lot. The police will gleefully give you a ticket if they catch you doing this in the wrong