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 pathyou 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 drivingI 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 limitin other words, we can derive
maximum cornering speed. For the following discussion, it will
be helpful for you to draw little backoftheenvelope
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 fastyour'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.
