Rolling Friction

What rolling friction is

When a body rolls on a surface — a wheel, a ball, a cylinder — there is a resistance to its motion called rolling friction:

$f_r = \mu_r N$

where $\mu_r$ is the coefficient of rolling friction and $N$ is the normal force.

Why rolling friction exists

A rolling body does not slide — the contact point is instantaneously at rest. So why is there any friction at all?

Rolling friction arises from deformation — both of the rolling body and the surface:

• The surface deforms slightly ahead of the contact region (compression)
• The surface springs back slightly behind the contact (extension)
• This deformation is not perfectly elastic — energy is lost as heat

The result is that the normal reaction from the surface is shifted slightly forward of the contact point, creating a torque that opposes the rolling motion.

Additionally, at the microscopic level, there is some adhesion and micro-slip at the contact region.

Rolling friction is much smaller than sliding friction

$\mu_r \ll \mu_k \ll \mu_s$

Typical values:

Compare with $\mu_k$ for rubber on concrete: $\approx 0.6$ — about 30–60 times larger.

This is precisely why wheels were invented. Rolling replaces sliding and reduces friction dramatically, allowing heavy loads to be moved with little effort.

Why wheels work

A wheel rolling without slipping has its contact point momentarily at rest relative to the ground. The friction at the contact point is static, not kinetic. Static friction can provide a large force (up to $\mu_s N$) without energy dissipation.

Rolling friction (the small resistance to rolling) is not due to this contact friction — it is due to the deformation losses described above.

Units and relation to normal force

Like all friction coefficients, $\mu_r$ is dimensionless. Rolling friction, like static and kinetic friction, is proportional to the normal force — pressing a wheel harder into the ground increases rolling resistance.