What is angular velocity?

Angular velocity (symbol ω, pronounced omega) is the rate at which an object rotates about an axis. It measures how many radians of angle are swept per second. The SI unit is rad/s. A full revolution equals 2π radians (approximately 6.2832 rad), so one revolution per second corresponds to ω = 2π rad/s ≈ 6.283 rad/s.

What is the formula for angular velocity?

The fundamental definition is ω = Δθ ÷ Δt, where Δθ is the angular displacement in radians and Δt is the time in seconds. For constant rotational speed expressed in revolutions per minute (rpm), the conversion is ω = 2π · n ÷ 60, where n is the speed in rpm. If you know the tangential (linear) speed v and the radius r, then ω = v ÷ r.

How do I convert rpm to rad/s?

Use the formula ω = 2π · n ÷ 60. For example, 3000 rpm gives ω = 2π × 3000 ÷ 60 = 314.16 rad/s. Select the "Convert RPM → rad/s" mode and enter your rpm value to get the instant conversion along with the period and frequency.

What is centripetal acceleration and how is it related to angular velocity?

Centripetal acceleration (a_c) is the inward acceleration that keeps an object on a circular path. Its relationship to angular velocity is a_c = ω² · r, where r is the radius. Equivalently, a_c = v² ÷ r where v = ω · r is the tangential speed. Select the centripetal mode, enter ω and r, and the calculator returns both a_c and v_t.

What is the difference between angular velocity and angular acceleration?

Angular velocity ω describes how fast something is rotating (rad/s). Angular acceleration α describes how quickly the rotation rate is changing (rad/s²), defined as α = Δω ÷ Δt. If a spinning wheel speeds from 2 rad/s to 10 rad/s in 4 s, its angular acceleration is (10 − 2) ÷ 4 = 2 rad/s². Use the angular acceleration mode to compute α, the angle swept and the average ω over the interval.

What are typical angular velocity values?

A clock's minute hand turns at 2π ÷ 3600 ≈ 0.00175 rad/s. A car engine at 3000 rpm runs at ≈ 314 rad/s. A domestic washing-machine drum at 1200 rpm spins at ≈ 126 rad/s. A CD at 500 rpm gives ≈ 52 rad/s. Earth rotates at ≈ 7.27 × 10⁻⁵ rad/s (one revolution per 24 hours).

What is the Angular Velocity Calculator?

Free angular velocity calculator. Solve for ω, Δθ, Δt, RPM conversions, centripetal acceleration, tangential velocity and angular acceleration. Runs entirely in your browser with full formula working shown. It runs free in your browser on Gera Tools, with nothing uploaded.

Angular Velocity Calculator

Name: Angular Velocity Calculator
Creator: Gera Tools
License: https://creativecommons.org/licenses/by/4.0/

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The Angular Velocity Calculator covers the full set of rotational-motion relationships you meet in A-level physics, university mechanics, and engineering design: the core definition ω = Δθ ÷ Δt, the link between RPM and rad/s, tangential and centripetal quantities, and angular acceleration. Every calculation is done live in your browser — nothing is uploaded or stored.

How it works

Angular velocity is defined as the angle swept divided by the time taken:

ω = Δθ ÷ Δt

where ω is in rad/s, Δθ is the angular displacement in radians, and Δt is the time in seconds. Because one full revolution is 2π radians, an object completing one revolution per second has ω = 2π ≈ 6.283 rad/s.

Unit conversions built in

The calculator automatically converts your result to rpm (revolutions per minute) using n = ω × 60 ÷ (2π), and to degrees per second using ω_deg = ω × 180 ÷ π. It also reports the period T = 2π ÷ ω (time for one full revolution) and frequency f = ω ÷ (2π) in Hz.

Tangential and centripetal quantities

A point at radius r from the rotation axis travels at tangential speed:

v = ω · r

and experiences centripetal acceleration directed inward:

a_c = ω² · r = v² ÷ r

These are the key bridge equations between rotational and translational (linear) motion, used in everything from gear design to orbital mechanics.

Angular acceleration

When the rotation rate changes, angular acceleration α (rad/s²) is:

α = Δω ÷ Δt

The angle swept during constant angular acceleration follows the rotational analogue of the kinematic suvat equations:

θ = ω_i · t + ½ · α · t²

This mirrors the linear kinematic formula s = u·t + ½·a·t² with every linear quantity replaced by its rotational counterpart.

Worked example

Problem: A motor spins up from rest to 3000 rpm in 5 seconds. Find: (a) the final angular velocity in rad/s; (b) the angular acceleration; (c) the angle swept during spin-up; (d) the centripetal acceleration at the rim of a disc of radius 0.12 m.

Step 1 — Convert 3000 rpm to rad/s:

ω_f = 2π × 3000 ÷ 60 = 314.16 rad/s

Step 2 — Angular acceleration (initial ω = 0):

α = (314.16 − 0) ÷ 5 = 62.83 rad/s²

Step 3 — Angle swept:

θ = 0 × 5 + ½ × 62.83 × 5² = 785.4 rad (≈ 125 complete revolutions)

Step 4 — Centripetal acceleration at the rim:

a_c = ω_f² × r = 314.16² × 0.12 = 11 845 m/s² (≈ 1 207 g)

Rim engineering clearly demands careful material selection at high rpm.

Formula reference

Quantity	Formula	Units
Angular velocity	ω = Δθ ÷ Δt	rad/s
From rpm	ω = 2π · n ÷ 60	rad/s
From linear speed	ω = v ÷ r	rad/s
Tangential speed	v = ω · r	m/s
Centripetal acceleration	a_c = ω² · r	m/s²
Angular acceleration	α = Δω ÷ Δt	rad/s²
Angle swept (const. α)	θ = ω_i·t + ½·α·t²	rad
Period	T = 2π ÷ ω	s
Frequency	f = ω ÷ 2π	Hz

All constants are the standard SI values used in A-level and university physics. The calculator rounds results to six significant figures.