Grand Prix Racing   The Science of Fast Pinewood Cars 
It is a relatively simple matter to determine static friction coefficients. It is the measure of the friction coefficient at the onset of motion. Once the car is moving, friction coefficients are refered to as kinetic. The static tread friction coefficient must be determined first using a single wheel or a set of wheels. Only then can the static axle friction coefficient of your car can be determined easily.
Some physicists (like the late nobel prize winner Richard P. Feynamn) believe that for most practical purposes, static friction coefficients are the same in value as their kinetic counterparts. All agree, that their values are not lower. So, the method presented here should yeild in the worst case, friction coefficients that are at least upper bounds on the kinetic values, and in the best case, the same.
To measure static tread friction, you will need:
The measurement is taken by placing the wheel in the middle of the ramp while it is flat on the ground then lifting one end of the ramp until the wheel just moves. It is very easy to derive the relationship between the ramp angle, O, and the static rolling friction coefficient, u.
The wheel begins to roll just when the forward part of the gravitational pull is equal to or a little greater than the opposing friction. At the point where these forces are equal we can write:
0 = mgsinO  umg
divide through by the weight of the wheel, mg, to get
u = sinO
Let H be the height of the ramp top edge when the wheel begins to roll, h be the thickness of the ramp edge and Lr be the length of the ramp surface. Then
sinO = (Hh)/Lr
To calculate u, all we have to do is measure H, h and Lr. We don't really need to know the angle!
Symbol  Value  Description 

Lr  __________  Length of ramp surface in inches 
h  __________  Height of ramp edge in inches 
H  __________  Height of ramp in inches when wheel starts to roll 
u  __________  Rolling drag coefficient u = (Hh)/Lr 
There is one more thing to note. If your wheel won't stand up on the ramp, put an axle through two of them and make your measurements that way. The equation for u says that the weight of the wheel doesn't matter! Only it's material matters. But that raises the issue of different coefficients for different materials, both wheels and ramps.

To measure static tread friction, you will need:
The measurement is taken by placing your car in the middle of the ramp while it is flat on the ground then lifting one end of the ramp until the car just moves. The relationship between the ramp angle, O, and the static axle friction coefficient, n, is more complicated than for the static rolling friction coefficient. Not only does the expression for total force involve the static rolling friction coefficient, but it depends on how many wheels are lifted and how much they weigh.
The car begins to roll just when the forward part of the gravitational pull is equal to or a little greater than the opposing friction. At the point where these forces are equal we can write:
0 = mgsinO  Dr for Dr the total wheel drag.
From the race model summary, the total drag on the car from axle and tread friction is:
Dr = (nfFfrf/Rf + 2uf(Ff/2+mfg) + nrFrrr/Rr + 2ur(Fr/2+mrg))cosO
When a front wheel is lifted, this becomes:
Dr = (nf(Ff+mfg)rf/Rf + uf((Ff+mfg)+mfg) + nrFrrr/Rr + 2ur(Fr/2+mrg))cosO note: the front tread friction remained the same: uf((Ff+mfg)+mfg) = 2uf(Ff/2+mfg)
When a front wheel and rear wheel is lifted, this becomes:
Dr = (nf(Ff+mfg)rf/Rf + uf((Ff+mfg)+mfg) + nr(Fr+mrg)rr/Rr + ur((Fr+mrg)+mrg))cosO note: the front and rear tread friction remained the same.
When two front wheels are lifted, this becomes:
Dr = nr(mg2mrg)rr/Rr + urmg)cosO note: mg is the total weight of the car.
To get the values for Ff and Fr, their defining equations appearing in the summary would have to be evaluated with zero centrifugal force. When added together, Ff+Fr is the total weight of the car minus the weight of the wheels touching the ramp:
Ff+Fr = mgjmfglmrg for j and k the number of front wheels and rear wheels touching the ramp (ie. not lifted).
For most cars, the front and rear wheels are the same. So let nf = nr, rf = rr, Rf = Rr, uf = ur, w = mf = mr and let j = j+k (from above). Then the drag force on the ramp after fixing the tread friction for j wheels touching becomes:
Dr = (n(Ff+Fr)r/R + 2u((Ff+Fr)/2+jwg/2))cosO and Ff+Fr = mgjwg
Which after plugging in the expression for Ff+Fr above becomes:
Dr = (n(mgjwg)r/R + 2u((mgjwg)/2+jwg/2))cosO
Simplify
Dr = (n(mgjwg)r/R + umg)cosO
At the point where the car begins to roll, we now have,
0 = mgsinO  (n(mgjwg)r/R + umg)cosO
Since we measured, u above, we can solve for n.
(n(mgjwg)r/R + umg)cosO = mgsinO
Divide by the cosine to get the tangent on the right side
n(mgjwg)r/R + umg = mgtanO
Multiply and divide through to isolate n, the static axle friction coefficient
n = (mg(u+tanO))R/r(mgjwg)

Let H be the height of the ramp top edge when your car begins to roll, h be the thickness of the ramp edge and Lr be the length of the ramp surface. Then
O = arcsin((Hh)/Lr)
So the expression for n becomes,
n = (mg(utan(arcsin((Hh)/Lr))))R/r(mgjwg)
To calculate n, all we have to do is measure H, h, Lr, R, r, the weight of the car and a wheel and decide how many wheels are touching the ramp.
Symbol  Value  Description 

Lr  __________  Length of ramp surface in inches 
h  __________  Height of ramp edge in inches 
H  __________  Height of ramp in inches when wheel starts to roll 
R  __________  Outer radius of a wheel in inches 
r  __________  Inner radius of a wheel in inches 
mg  __________  Weight of car in ounces 
wg  __________  Weight of a wheel in ounces 
j  __________  Number of wheels touching the ramp 
n  __________  Axle drag coefficient n = (mg(utan(arcsin((Hh)/Lr))))R/r(mgjwg) 
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Grand Prix Racing   The Science of Fast Pinewood Cars 
Copyright © 1997, 2004 by Michael Lastufka, All rights reserved worldwide. 