Rolling resistance coefficient  [  [ edit
](/w/index.php?title=Rolling_resistance&action=edit&section=3 "Edit section:
Rolling resistance coefficient") ]

The "rolling resistance coefficient" is defined by the following equation:
[6]

F  =  C  r  r  N  {\displaystyle \ F=C_{rr}N}

![{\\displaystyle \\
F=C_{rr}N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40774ba595e9340365a0f3e0367af9a1fdfe872d)

where

* F  {\displaystyle F}  ![{\\displaystyle F}](https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57) is the rolling resistance force (shown as  R  {\displaystyle R}  ![{\\displaystyle R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33) in figure 1),
* C  r  r  {\displaystyle C_{rr}}  ![{\\displaystyle C_{rr}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76) is the dimensionless **rolling resistance coefficient** or **coefficient of rolling friction** ( **CRF** ), and
* N  {\displaystyle N}  ![{\\displaystyle N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3) is the [ normal force ](/wiki/Normal_force "Normal force") , the force perpendicular to the surface on which the wheel is rolling.

C  r  r  {\displaystyle C_{rr}}  ![{\\displaystyle
C_{rr}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76)
is the force needed to push (or tow) a wheeled vehicle forward (at constant
speed on a level surface, or zero grade, with zero air resistance) per unit
force of weight. It is assumed that all wheels are the same and bear identical
weight. Thus:  C  r  r  =  0.01  {\displaystyle \ C_{rr}=0.01}
![{\\displaystyle \\
C_{rr}=0.01}](https://wikimedia.org/api/rest_v1/media/math/render/svg/715f969199ffbdbf56c2a4858864537a03ca3a94)
means that it would only take 0.01 pounds to tow a vehicle weighing one pound.
For a 1000-pound vehicle, it would take 1000 times more tow force, i.e. 10
pounds. One could say that  C  r  r  {\displaystyle C_{rr}}  ![{\\displaystyle
C_{rr}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76)
is in lb(tow-force)/lb(vehicle weight). Since this lb/lb is force divided by
force,  C  r  r  {\displaystyle C_{rr}}  ![{\\displaystyle
C_{rr}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76)
is dimensionless. Multiply it by 100 and you get the percent (%) of the weight
of the vehicle required to maintain slow steady speed.  C  r  r
{\displaystyle C_{rr}}  ![{\\displaystyle
C_{rr}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76)
is often multiplied by 1000 to get the parts per thousand, which is the same
as kilograms (kg force) per metric ton (tonne = 1000 kg ),  [15]  which is the
same as pounds of resistance per 1000 pounds of load or Newtons/kilo-Newton,
etc. For the US railroads, lb/ton has been traditionally used; this is just
2000  C  r  r  {\displaystyle 2000C_{rr}}  ![{\\displaystyle
2000C_{rr}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1eaa901806c06576187f00cadf2b7e3bd4cb775f)
. Thus, they are all just measures of resistance per unit vehicle weight.
While they are all "specific resistances", sometimes they are just called
"resistance" although they are really a coefficient (ratio)or a multiple
thereof. If using pounds or kilograms as force units, mass is equal to weight
(in earth's gravity a kilogram a mass weighs a kilogram and exerts a kilogram
of force) so one could claim that  C  r  r  {\displaystyle C_{rr}}
![{\\displaystyle
C_{rr}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76)
is also the force per unit mass in such units. The SI system would use N/tonne
(N/T, N/t), which is  1000  g  C  r  r  {\displaystyle 1000gC_{rr}}
![{\\displaystyle
1000gC_{rr}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d4d63ab69873f3ca7a3e7726be3a0fcbcea07fb)
and is force per unit mass, where _g_ is the acceleration of gravity in SI
units (meters per second square).  [16]

The above shows resistance proportional to  C  r  r  {\displaystyle C_{rr}}
![{\\displaystyle
C_{rr}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76)
but does not explicitly show any variation with speed,  loads  ,  torque  ,
surface roughness,  diameter  , tire inflation/wear, etc., because  C  r  r
{\displaystyle C_{rr}}  ![{\\displaystyle
C_{rr}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76)
itself varies with those factors. It might seem from the above definition of
C  r  r  {\displaystyle C_{rr}}  ![{\\displaystyle
C_{rr}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76)
that the rolling resistance is directly proportional to vehicle weight but  it
is not  .