8.8 Rolling Resistance8.8 Rolling Resistance

� If a rigid cylinder of weight W rolls at constant velocity along a rigid surface, the normal force exerted by the surface on the cylinder acts at the tangent point of contactthe tangent point of contact

� Under these conditions, provided the cylinder does not encounter frictional resistance from the air, motion will continue indefinitely

� No materials are perfectly rigid and the reaction of the surface on the cylinder consists of a distribution of normal pressure

8.8 Rolling Resistance8.8 Rolling Resistance

� Consider the cylinder to be made of a very hard material and the surface on which it rolls to be relatively softer

� Due to its weight, the cylinder � Due to its weight, the cylinder compresses the surface underneath it

� As the cylinder rolls, the surface material in front of the cylinder retards the motion since it is being deformed whereas the material in the rear is restored from the deformed state and tends to push the cylinder forward

8.8 Rolling Resistance8.8 Rolling Resistance

� The normal pressures acting on the cylinder in this manner are represented by their resultant forces Nd and Nr

The magnitude of the force of � The magnitude of the force of deformation Nd and its horizontal component is always greater than that of restoration Nr and consequently, a horizontal driving force P must be applied to maintain the motion

8.8 Rolling Resistance8.8 Rolling Resistance

� Rolling resistance is caused primarily by this effect although the result of surface adhesion and relative micro-sliding between the surfaces of sliding between the surfaces of contact

� Actual force P needed to overcome these effects is difficult to determine, so we consider the resultant of the entire normal pressure acting on the cylinder

N = Nd + Nr

8.8 Rolling Resistance8.8 Rolling Resistance

� Force P acts at an angle θ with the vertical

� To keep the cylinder in equilibrium, rolling at constant rate, N must beconcurrent with the driving force P concurrent with the driving force P

and the weight W

� Summation of moment about A,

Wa = P (r cosθ)

Wa ≈ Pr

P ≈ (Wa)/r

8.8 Rolling Resistance8.8 Rolling Resistance

Example 8.12

A 10kg steel wheel has a radius

of 100mm and rest on an

inclined plans made of wood. If θinclined plans made of wood. If θ

is increased so that the wheel

begins to roll down the incline

with constant velocity when θ =

1.2°, determine the coefficient of

rolling resistance.

8.8 Rolling Resistance8.8 Rolling Resistance

Solutions

� FBD of the wheel

� Wheel has impending motion

� Normal reaction N acts at point A � Normal reaction N acts at point A defined by dimension a

� Summing moments about point A

� Solving

mma

mmNaN

M A

09.2

0)100(2.1sin81.9)(2.1cos81.9

;0

=

=−

=∑

oo