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 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
newsletter).
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)
Acceleration
Radius (Feet)
(Gees)
50.00
100.0
150.0
200.0
500.0
0.25
13.66
19.31
23.66
27.32
43.19
0.50
19.31
27.32
33.45
38.63
61.08
0.75
23.66
33.45
40.97
47.31
74.81
1.00
27.32
38.63
47.31
54.63
86.38
1.25
30.54
43.19
52.90
61.08
96.57
1.50
33.45
47.31
57.94
66.91
105.79
1.75
36.13
51.10
62.59
72.27
114.27
2.00
38.63
54.63
66.91
77.26
122.16
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 (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 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 places.
