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 external force.
When you turn the steering wheel, you are trying to get the front tires 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 tires 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 center 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 center of the circle. This force is
called *centripetal*, from the Greek for ``center 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 ``center 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 in a circle
of radius . 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 newsletter).
Consider a very short interval of time, far less than a second. Call it ( stands for ``delta,'' a Greek letter
mathematicians use as shorthand for ``tiny increment''). In time , let us say we go forward a distance and sideways a distance . The forward component of
the velocity of the car is approximately . At the beginning of the time interval , the car has no sideways
velocity. At the end, it has sideways velocity . In the time , the car has thus had a change in sideways
velocity of .
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 related to , the radius of the circle? If we go forward
by a fraction of the
radius of the circle, we must go sideways by exactly the same fraction of to stay on the circle.
This means that . The
fraction is,
however, nothing but .
By this reasoning, we get the relation We can substitute this expression for into the expression for , and remembering that , 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 tires 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-your'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 for , the velocity. The conversion factor 15/22
converts from feet
per second to miles per hour, and 32.1 converts from gees to feet per second squared. We
covered these conversion factors in part 3 of this series.
For autocrossing, the columns for 50 and 100 feet and the row for 1.00 are most germane. The
table tells us that to achieve 1.00 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, 1.25 is required, which is probably not within
the capability of an autocross tire 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 maneuver 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 (Ayrton Senna,
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 colors 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 places. |