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cylinder retards the motion since it is being deformed, whereas the
material in the rear is restored from the deformed state and therefore
tends to push the cylinder forward. The normal pressures acting on the
cylinder in this manner are represented in Fig. 1-67,b by their resultant
forces N and N . Because the magnitude of the force of deformation
d
r
N , and its horizontal component is always greater than that of
d
restoration, N , and consequently a horizontal driving force P must
r
be applied to the cylinder to maintain the motion, Fig. 1-67,b.
Rolling resistance is caused primarily by this effect, although it
is also, to a lesser degree, the result of surface adhesion and relative
microsliding between the surfaces of contact. Because the actual force
P needed to overcome these effects is difficult to determine, a
simplified method will be developed here to explain one way
engineers have analyzed this phenomenon. To do this, we will
consider the resultant of the entire normal pressure, N = N d + N ,
r
acting on the cylinder, Fig. 1-67,c. As shown in Fig. 1-67,d, this force
acts at an angle with the vertical. To keep the cylinder in equilibrium,
i.e., rolling at a constant rate, it is necessary that N be concurrent with
the driving force P and the weight W. Summing moments about point
A gives Wa = P ( cos )r θ . Since the deformations are generally very
small in relation to the cylinder’s radius, cosθ = 1; hence,

Wa ≈ Pr
or
Wa
P ≈ 1-54
r
The distance a is termed the coefficient of rolling resistance,
which has the dimension of length. For instance, a ≈ 0.5 mm for a
wheel rolling on a rail, both of which are made of mild steel. For
hardened steel ball bearings on steel, a ≈ 0.1 mm. Experimentally,
though, this factor is difficult to measure, since it depends on such
parameters as the rate of rotation of the cylinder, the elastic properties
of the contacting surfaces, and the surface finish. For this reason, little
reliance is placed on the data for determining a. The analysis
presented here does, however, indicate why a heavy load (W) offers
greater resistance to motion (P) than a light load under the same

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