Angular Velocity Calculator – Calculate ω from RPM, Linear Velocity & More

  • Adam
  • November 27, 2025

Calculate angular velocity (ω) instantly from RPM, angular displacement, linear velocity, frequency, or period. Free calculator with step-by-step solutions, unit conversions, and comprehensive formulas in rad/s.

Angular Velocity Calculator - Calculate ω in Rad/s from RPM & More

An angular velocity calculator determines the rate of rotation of an object around an axis, measured in radians per second (rad/s). Angular velocity is fundamental to understanding circular motion, rotational dynamics, and mechanical systems, with applications ranging from motor speed calculations to planetary motion analysis.

Calculate Angular Velocity

Result:

What is Angular Velocity?

Angular velocity is a vector quantity that describes how fast an object rotates or revolves around an axis. It measures the rate of change of angular position with respect to time, representing the speed of rotation in radians per second. Unlike linear velocity which measures straight-line motion, angular velocity quantifies rotational motion and is fundamental to understanding circular dynamics, rotating machinery, and orbital mechanics.

Key Characteristics of Angular Velocity:

  • Vector quantity: Has both magnitude (speed of rotation) and direction (axis orientation following right-hand rule)
  • Units: Radians per second (rad/s) in SI, also expressed as degrees/second or RPM in practical applications
  • Symbol: ω (Greek letter omega)
  • Constant for rigid bodies: Every point on a rigid rotating body has the same angular velocity
  • Related to linear velocity: Connected through the relationship v = ωr
  • Dimensional formula: [T⁻¹] (per time)

Angular Velocity Formula

Angular velocity can be calculated using different formulas depending on which parameters are known. The fundamental relationships connect angular velocity to angular displacement, time, frequency, RPM, and linear velocity.

Basic Angular Velocity Formula

ω = Δθ / Δt

Where:

  • ω = angular velocity (rad/s)
  • Δθ = change in angular displacement (radians)
  • Δt = change in time (seconds)

Angular Velocity from RPM

ω = 2πn / 60

or equivalently

ω = πn / 30

Where:

  • ω = angular velocity (rad/s)
  • n = rotational speed (RPM, revolutions per minute)
  • 2π = radians per revolution
  • 60 = seconds per minute

Angular Velocity from Linear Velocity

ω = v / r

or equivalently

v = ωr

Where:

  • ω = angular velocity (rad/s)
  • v = linear/tangential velocity (m/s)
  • r = radius or distance from rotation axis (m)

Angular Velocity from Frequency and Period

From Frequency:

ω = 2πf

From Period:

ω = 2π / T

Where:

  • f = frequency (Hz or cycles per second)
  • T = period (seconds per cycle)
  • Note: f = 1/T (frequency and period are reciprocals)

Related Formulas

Angular Acceleration:

α = dω/dt = Δω/Δt

Rotational Kinetic Energy:

Erot = ½Iω²

Angular Momentum:

L = Iω

Where I is moment of inertia

How to Calculate Angular Velocity

Calculating angular velocity requires identifying which parameters are known and selecting the appropriate formula. Follow these systematic steps:

  1. Identify known parameters: Determine what information you have—angular displacement and time, RPM, linear velocity and radius, frequency, or period.
  2. Select the appropriate formula: Choose the equation that matches your known variables from the formulas above.
  3. Convert units if necessary: Ensure angles are in radians, time in seconds, and distances in meters for SI calculations.
  4. Substitute values: Plug the known values into the selected formula.
  5. Calculate the result: Perform the mathematical operations to determine angular velocity in rad/s.
  6. Convert to desired units: If needed, convert rad/s to degrees/second or RPM for practical applications.
  7. Verify reasonableness: Check that the result makes physical sense for the system being analyzed.

Calculation Example 1: From RPM

Problem: A motor shaft rotates at 1800 RPM. Calculate the angular velocity in rad/s.

Solution:

Given: n = 1800 RPM

Using formula: ω = 2πn / 60

ω = 2π × 1800 / 60

ω = 2π × 30

ω = 60π rad/s

ω = 188.50 rad/s

Alternative calculation:

Using ω = πn/30

ω = π × 1800 / 30 = 60π = 188.50 rad/s ✓

Calculation Example 2: From Linear Velocity

Problem: A point on the edge of a disk moves with velocity 15 m/s. The disk has radius 0.5 m. Find the angular velocity.

Solution:

Given: v = 15 m/s, r = 0.5 m

Using formula: ω = v / r

ω = 15 / 0.5

ω = 30 rad/s

Additional information:

Converting to RPM: n = 60ω/2π = 60 × 30 / (2π) = 286.5 RPM

Calculation Example 3: From Angular Displacement

Problem: A wheel rotates through 720 degrees in 4 seconds. Calculate the angular velocity.

Solution:

Step 1: Convert degrees to radians

θ = 720° × π/180 = 4π radians = 12.566 rad

Step 2: Calculate angular velocity

Given: θ = 4π rad, t = 4 s

ω = θ / t = 4π / 4 = π rad/s

ω = 3.142 rad/s

This corresponds to:

• 180 degrees per second

• 0.5 revolutions per second

• 30 RPM

Angular Velocity vs Linear Velocity

Understanding the relationship and differences between angular and linear velocity is essential for analyzing rotational motion:

Characteristic Angular Velocity (ω) Linear Velocity (v)
Definition Rate of change of angular position Rate of change of linear position
Units rad/s, deg/s, RPM m/s, km/h, mph
Formula ω = Δθ/Δt v = Δs/Δt
Applies to Rotational motion Translational motion
Relationship v = ωr ω = v/r
Same for rigid body? Yes (all points same ω) No (depends on distance from axis)

Relationship Between Angular and Linear Velocity

For circular motion, angular and linear velocity are directly related through the radius:

v = ωr

This fundamental equation shows that:

  • Points farther from the rotation axis move faster linearly but have the same angular velocity
  • Doubling the radius doubles the linear velocity for the same angular velocity
  • This explains why the outer edge of a spinning disk moves faster than points near the center

Unit Conversions for Angular Velocity

Angular velocity can be expressed in various units depending on the application. Here are common conversions:

Conversion Table

From To Multiply By
RPM rad/s π/30 ≈ 0.10472
rad/s RPM 30/π ≈ 9.5493
rad/s deg/s 180/π ≈ 57.2958
deg/s rad/s π/180 ≈ 0.01745
rev/s rad/s 2π ≈ 6.2832
rad/s rev/s 1/(2π) ≈ 0.1592

Applications of Angular Velocity

Angular velocity is a critical parameter in numerous engineering, scientific, and practical applications:

  • Mechanical Engineering: Designing motors, turbines, gears, flywheels, and analyzing rotating machinery performance
  • Automotive: Engine crankshaft speed (RPM), wheel rotation, transmission analysis, and vehicle dynamics
  • Aerospace: Propeller and rotor speeds, spacecraft attitude control, gyroscope operation, and satellite rotation
  • Manufacturing: Lathe and milling machine speeds, CNC machining parameters, and centrifuge operations
  • Astronomy: Planetary rotation rates, orbital motion, stellar spin, and galactic rotation curves
  • Sports: Ball spin rates (golf, tennis, baseball), figure skating rotations, and diving maneuvers
  • Power Generation: Generator and turbine speeds, wind turbine RPM, and hydroelectric systems
  • Consumer Electronics: Hard disk drive speeds, cooling fan RPM, and motor-driven appliances

Frequently Asked Questions

What is the formula for angular velocity?
The basic formula for angular velocity is ω = Δθ/Δt, where ω is angular velocity in rad/s, Δθ is the change in angular displacement in radians, and Δt is time in seconds. Other common formulas include ω = 2πn/60 for converting from RPM, ω = v/r for relating to linear velocity, and ω = 2πf for converting from frequency. The choice of formula depends on which parameters are known.
How do you convert RPM to angular velocity?
To convert RPM to angular velocity in rad/s, use the formula ω = 2πn/60 or ω = πn/30, where n is RPM. For example, 300 RPM converts to ω = 2π × 300/60 = 10π = 31.42 rad/s. This conversion accounts for 2π radians per revolution and 60 seconds per minute. Alternatively, you can multiply RPM by 0.10472 (which equals π/30) to get rad/s directly.
What is the unit of angular velocity?
The SI unit of angular velocity is radians per second (rad/s). Other commonly used units include degrees per second (deg/s), revolutions per minute (RPM), and revolutions per second (rev/s or rps). In physics and engineering, rad/s is preferred because it simplifies equations. The dimensional formula is [T⁻¹] (per time). Since radians are dimensionless, angular velocity has the same dimensions as frequency.
What is the relationship between angular velocity and linear velocity?
Angular velocity and linear velocity are related by the equation v = ωr, where v is linear velocity, ω is angular velocity, and r is the radius or distance from the rotation axis. This means linear velocity equals angular velocity multiplied by radius. For example, if ω = 10 rad/s and r = 2 m, then v = 20 m/s. Points farther from the axis have higher linear velocity but the same angular velocity in rigid body rotation.
Is angular velocity the same as angular frequency?
Angular velocity and angular frequency are numerically identical and both measured in rad/s, but they have slightly different conceptual meanings. Angular velocity typically refers to the rotation rate of a physical object (like a spinning wheel), while angular frequency refers to the temporal oscillation rate of a wave or periodic phenomenon (like in simple harmonic motion). The formula ω = 2πf applies to both. In many contexts, the terms are used interchangeably.
How do you calculate angular velocity from angular displacement?
To calculate angular velocity from angular displacement, divide the angular displacement by the time taken: ω = θ/t. Ensure the angle is in radians and time is in seconds for the result in rad/s. For example, if an object rotates through 3π radians in 5 seconds, then ω = 3π/5 = 1.885 rad/s. If the angle is given in degrees, first convert to radians by multiplying by π/180.
What is the difference between angular velocity and angular acceleration?
Angular velocity (ω) is the rate of change of angular position, measured in rad/s, while angular acceleration (α) is the rate of change of angular velocity, measured in rad/s². Angular velocity tells you how fast something is rotating; angular acceleration tells you how quickly the rotation rate is changing. They are related by α = dω/dt or α = Δω/Δt. Constant angular velocity means zero angular acceleration.
Can angular velocity be negative?
Yes, angular velocity can be negative. The sign indicates the direction of rotation according to the right-hand rule. By convention, counterclockwise rotation (when viewed from above) is positive, and clockwise rotation is negative. For example, ω = -20 rad/s means clockwise rotation at 20 rad/s. The magnitude (absolute value) represents the speed of rotation, while the sign indicates direction. Some problems use different sign conventions, so always check the coordinate system definition.
How do you find angular velocity from period?
To find angular velocity from period, use ω = 2π/T, where T is the period (time for one complete rotation) in seconds. For example, if an object completes one rotation every 4 seconds, then ω = 2π/4 = π/2 = 1.571 rad/s. This formula comes from the fact that one complete rotation is 2π radians, so angular velocity equals angle per time: ω = 2π radians / T seconds. Period and frequency are reciprocals: T = 1/f.
Why is angular velocity constant for all points on a rigid body?
Angular velocity is constant for all points on a rigid body because every point rotates through the same angle in the same time interval. While linear velocities differ (points farther from the axis move faster), the angular displacement Δθ is identical for all points. Since ω = Δθ/Δt and both Δθ and Δt are the same everywhere on the rigid body, ω must be constant. This is a fundamental property of rigid body rotation—the body rotates as a single unit without deformation.

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