Rolling Without Slipping

Equivalent forms

a = R \alpha

s = R \theta

A simple algebraic constraint that ties translation and rotation together — the entire reason wheels work.

Symbol	Name	SI unit	Dimension	Range
$v$	Translational speedoutput Speed of wheel center	m/s	LT^-1	0 – 200
$R$	Radius Wheel radius	m	L	0.05 – 2
$omega$	Angular speed Wheel spin rate	rad/s	T^-1	0 – 200

Unit systems

Symbol	SI	CGS	Imperial
$v$	m/s	cm/s	ft/s
$R$	m	cm	ft
$omega$	rad/s	rad/s	rad/s

Where it holds

Rigid wheel and ground, sufficient static friction at contact; breaks down for deformable contacts, low-friction surfaces, or sudden braking/accel.

Dimensional analysis

v \to [LT^{-1}]

R\cdot \omega \to [L]\cdot [T^{-1}] = [LT^{-1}]

m/s on both sides.

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Discovery

Leonhard Euler · 1758

Euler systematized rigid-body kinematics, distinguishing rolling from sliding contact — a foundation for everything from gears to railway dynamics.

Try this

A car wheel of radius 0.3 m spins at 100 rad/s — how fast is the car moving?

Find the translational speed of a wheel rolling without slipping at angular velocity ω with radius R.

Research status: stable

Real-world applications

Vehicle drivetrains and gearing
Conveyor belts and pulleys
Rolling-ball billiard physics
Train wheel-rail dynamics

Common misconceptions

The bottom of the wheel moves at v — it momentarily moves at 0 (relative to ground)
Rolling needs no friction — it needs STATIC friction to maintain the constraint
All wheels of the same R roll the same way — moment of inertia matters for energy distribution

Experimental verification

Verified daily by every car odometer (which integrates

R\omega )

; tire-mark studies in accident reconstruction; rolling-friction experiments.

Derivation

The point on the wheel in contact with the ground has zero instantaneous velocity (no-slip).

The center of the wheel translates at v; the rim point at the bottom moves at

v - R\omega in

the ground frame.

Setting this to zero:

v = R\omega

.

Differentiating yields

a = R\alpha

.

Limiting cases

\omega = 0

⟶

v = 0

Wheel stationary — no rolling.

Slipping⟶

v \neq R\omega

Constraint broken; treat translation and rotation independently.

Pure spin⟶

v = 0

,

\omega \neq 0

Wheel spinning in place (peeling out) — also breaks no-slip.

What if…

What if the surface is icy?

Static friction max may be too small; wheel slips, $v \neq R\omega$ .

What about acceleration?

Differentiate: $a = R\alpha$ . Linear and angular accelerations are tied.

What if I look at energy?

KE_total $= \tfrac{1}{2} m v^{2} + \tfrac{1}{2} I_cm \omega ^{2} = \tfrac{1}{2}(m + I_cm/R^{2}) v^{2}$ — heavier rims (more I) means more energy at the same v.

1

Car wheel

Given ·

R:: 0.3
omega:: 100

Find · v

Steps

Apply $v = R\omega$
$v = 0.3 \times 100 = 30\,\mathrm{m/s}$
Convert: $30\,\mathrm{m/s} \times 3.6 \approx 108\,\mathrm{km/h}$

Result ·

v = 0.3 \times 100 = 30\,\mathrm{m/s} (108\,\mathrm{km/h})

2

Bicycle wheel

Given ·

R:: 0.35
omega:: 14.3

Find · v

Steps

$v = 0.35 \times 14.3$
$v \approx 5\,\mathrm{m/s}$
Cyclist's casual pace

Result ·

v \approx 5\,\mathrm{m/s} (18\,\mathrm{km/h})

Rotational Kinetic Energy→Kinetic Friction Force→Torque→Moment of Inertia (Point Mass)→