Angular Velocity Calculator

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

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