Linear Velocity Formula: v = ωr for Rotating Objects

The linear velocity formula is $v = \omega r$ : multiply the angular velocity ω, in radians per second, by the distance r from the axis of rotation. The answer is how fast a point on a spinning wheel, a fan blade or a merry go round is actually moving along its circular path, in metres per second.

This value is also called the tangential velocity, because its direction at any moment lies along the tangent line of the circle. Every point on a rigid turning body shares the same ω, but points farther out have larger r and so move faster. The tip of a ceiling fan blade races while the hub barely moves.

Quick facts

Formula
$v = \omega r$
Using the period
$v = \frac{2\pi r}{T}$
Using rpm
$\omega = \frac{2\pi N}{60}$ with N in rev/min
SI unit
m/s
Direction
Along the tangent to the circle
Rule for the angle
ω must be in rad/s, not degrees

Linear velocity formula and tangential velocity: the symbols

Terms in v = ωr
Symbol	Meaning	SI unit
$v$	Linear or tangential velocity of the point	m/s
$\omega$	Angular velocity, angle turned per second	rad/s
$r$	Distance of the point from the axis	m

The formula follows from the arc length rule $s = r\theta$ . Divide both sides by time: $\frac{s}{t} = r\frac{\theta}{t}$ , which is $v = r\omega$ . It only works with θ in radians, which is why ω must be in rad/s.

Worked examples

Example 1: a bicycle wheel

A bicycle wheel of radius 0.30 m turns at 10 rad/s. Find the speed of a point on the tyre.

$v = 10 \times 0.30 = 3.0\ \text{m/s}$

Example 2: a washing machine drum, from rpm

During the spin cycle a drum of radius 0.25 m turns at 900 revolutions per minute. How fast is the clothing pressed against its wall moving?

$\omega = \frac{2\pi \times 900}{60} = 30\pi \approx 94.2\ \text{rad/s}$

$v = 94.2 \times 0.25 \approx 23.6\ \text{m/s}$

That is about 85 km/h, which is why the water is flung out through the holes.

Example 3: a child on a roundabout

A playground roundabout makes one full turn every 10 s. A child sits 2.5 m from the centre. Find the child's linear velocity.

$\omega = \frac{2\pi}{T} = \frac{2\pi}{10} \approx 0.628\ \text{rad/s}$

$v = 0.628 \times 2.5 \approx 1.57\ \text{m/s}$

Check with $v = \frac{2\pi r}{T} = \frac{2\pi \times 2.5}{10} \approx 1.57$ m/s.

A friend sitting 1.25 m from the centre turns at the same ω but moves at half the speed, about 0.79 m/s.

Linear velocity in terms of period and frequency

In one turn, a point travels the circumference $2\pi r$ in one period T. So $v = \frac{2\pi r}{T}$ . Since frequency f = 1/T, you can also write $v = 2\pi r f$ . All three forms are the same formula. Pick the one that matches the data in the question: rad/s, a period in seconds, or a rate in revolutions per second.

The angular velocity formula page covers ω itself, and the period formula page covers T.

Why points farther out move faster

On a turning disc, every point sweeps the same angle each second. A point near the rim travels a bigger circle in that time, so it must move faster. Double the radius and the linear velocity doubles. That is why the outer horses on a carousel feel faster, and why the tip of a cricket bat moves faster than the handle during a swing.

Even at constant ω the direction of v keeps changing, so the point is accelerating toward the centre. That acceleration is v²/r, or ω²r, explained on the centripetal acceleration formula page.

Common mistakes

ω in rpm or degrees per second. Convert to rad/s first.
Diameter for radius. If a wheel is 0.60 m across, r = 0.30 m.
Thinking v is the same everywhere. It grows with r; only ω is shared.
Forgetting direction. The velocity is tangent to the circle, not toward the centre.

This page belongs with the straight line results on the kinematics formulas hub and the general velocity formula.

Frequently asked questions

What is the formula of linear velocity?

v = ωr, where ω is the angular velocity in rad/s and r is the radius in metres. You can also write v = 2πr / T.

Is linear velocity the same as tangential velocity?

For circular motion, yes. Both mean the velocity of a point along its path, which is tangent to the circle at every moment.

What is the unit of linear velocity?

It is measured in m/s. Radians carry no dimension, so multiplying rad/s by metres leaves plain metres per second.

How do you convert rpm to linear velocity?

First convert rpm to rad/s using ω = 2πN / 60, then multiply by the radius: v = ωr.

How does linear velocity differ from angular velocity?

Angular velocity is the angle turned per second and is the same for all points on a rigid body. Linear velocity is the distance moved per second and grows with distance from the axis.