Physical formulae  [  [ edit
](/w/index.php?title=Rolling_resistance&action=edit&section=5 "Edit section:
Physical formulae") ]

The coefficient of rolling resistance for a slow rigid wheel on a perfectly
elastic surface, not adjusted for velocity, can be calculated by  [18]  [ _[
citation needed  ](/wiki/Wikipedia:Citation_needed "Wikipedia:Citation
needed") _ ]

C  r  r  =  z  /  d  {\displaystyle C_{rr}={\sqrt {z/d}}}

![{\\displaystyle C_{rr}={\\sqrt
{z/d}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57b3c71bdeb68ae2f1f02894c090b21ee0a10513)

where

* z  {\displaystyle z}  ![{\\displaystyle z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98) is the sinkage depth
* d  {\displaystyle d}  ![{\\displaystyle d}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab) is the diameter of the rigid wheel

The empirical formula for  C  r  r  {\displaystyle C_{rr}}  ![{\\displaystyle
C_{rr}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76)
for cast iron mine car wheels on steel rails is:  [19]

C  r  r  =  0.0048  (  18  /  D  )  1  2  (  100  /  W  )  1  4  =  0.0643988
W  D  2  4  {\displaystyle C_{rr}=0.0048(18/D)^{\frac {1}{2}}(100/W)^{\frac
{1}{4}}={\frac {0.0643988}{\sqrt[{4}]{WD^{2}}}}}

![{\\displaystyle C_{rr}=0.0048\(18/D\)^{\\frac {1}{2}}\(100/W\)^{\\frac
{1}{4}}={\\frac
{0.0643988}{\\sqrt\[{4}\]{WD^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcdf0492d842147e528ef3ff457a3c7fb80b175d)

where

* D  {\displaystyle D}  ![{\\displaystyle D}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6) is the wheel diameter in inches
* W  {\displaystyle W}  ![{\\displaystyle W}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7) is the load on the wheel in pounds-force

As an alternative to using  C  r  r  {\displaystyle C_{rr}}  ![{\\displaystyle
C_{rr}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76)
one can use  b  {\displaystyle b}  ![{\\displaystyle
b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3)
, which is a different **rolling resistance coefficient** or **coefficient of
rolling friction** with dimension of length. It is defined by the following
formula:  [3]

F  =  N  b  r  {\displaystyle F={\frac {Nb}{r}}}

![{\\displaystyle F={\\frac
{Nb}{r}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f489c4b79d6e2d6b06536939ae05508303a93a55)

where

* F  {\displaystyle F}  ![{\\displaystyle F}](https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57) is the rolling resistance force (shown in figure 1),
* r  {\displaystyle r}  ![{\\displaystyle r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538) is the wheel radius,
* b  {\displaystyle b}  ![{\\displaystyle b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3) is the **rolling resistance coefficient** or **coefficient of rolling friction** with dimension of length, and
* N  {\displaystyle N}  ![{\\displaystyle N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3) is the normal force (equal to _W_ , not _R_ , as shown in figure 1).

The above equation, where resistance is inversely proportional to radius  r
{\displaystyle r}  ![{\\displaystyle
r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
seems to be based on the discredited "Coulomb's law" (Neither Coulomb's
inverse square law nor Coulomb's law of friction)  [ _[ citation needed
](/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed") _ ]  . See
dependence on diameter  . Equating this equation with the force per the
rolling resistance coefficient  , and solving for  b  {\displaystyle b}
![{\\displaystyle
b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3)
, gives  b  {\displaystyle b}  ![{\\displaystyle
b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3)
=  C  r  r  r  {\displaystyle C_{rr}r}  ![{\\displaystyle
C_{rr}r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/532a925f07b8de83675b265c5d746f4c39b8442b)
. Therefore, if a source gives rolling resistance coefficient (  C  r  r
{\displaystyle C_{rr}}  ![{\\displaystyle
C_{rr}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76)
) as a dimensionless coefficient, it can be converted to  b  {\displaystyle b}
![{\\displaystyle
b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3)
, having units of length, by multiplying  C  r  r  {\displaystyle C_{rr}}
![{\\displaystyle
C_{rr}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76)
by wheel radius  r  {\displaystyle r}  ![{\\displaystyle
r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
.