Productive ToolboxAngular Velocity Calculator
Calculate angular velocity in rad/s, deg/s, rev/s, and RPM from displacement, linear velocity, RPM, frequency, or period. Free online mechanical calculator with real-time results and unit conversions.
Angular Velocity Calculator
Choose a formula mode and enter values to instantly calculate angular velocity in rad/s, deg/s, rev/s, and RPM. Supports displacement, linear velocity, RPM, frequency, and period inputs.
Angular Velocity
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What is Angular Velocity?
Angular velocity (ω) is the rate at which an object rotates around an axis. It describes how fast the angular position changes over time and is the rotational equivalent of linear velocity in circular motion.
The standard SI unit is radians per second (rad/s). Other common units include degrees per second (deg/s), revolutions per second (rev/s), and revolutions per minute (RPM). This calculator supports all five input methods — displacement, linear velocity, RPM, frequency, and period — and converts results across all output units instantly.
Angular velocity is fundamental to mechanical engineering, robotics, automotive design, electrical machinery, and physics education. Any rotating system — from electric motors to planetary orbits — is characterized by its angular velocity.
How to Use the Angular Velocity Calculator
Step-by-Step Guide
- 1Select a formula mode from the dropdown (e.g. RPM Conversion)
- 2Enter the required input value(s) with appropriate units
- 3The angular velocity updates instantly in rad/s, deg/s, rev/s, and RPM
- 4Change the primary output unit in Settings to match your need
- 5Use presets to quickly load common engineering scenarios
- 6Click Copy Result or Export TXT to share or record your calculation
Key Features
- ✓5 formula modes — displacement, linear, RPM, frequency, period
- ✓Real-time calculation as you type
- ✓Instant unit conversion to all 4 output units
- ✓Step-by-step calculation breakdown
- ✓Live formula display with your actual values
- ✓Calculation history that persists between visits
- ✓Export results as a TXT file
- ✓Quick presets for common engineering scenarios
- ✓Selectable decimal precision (2–8 places)
Angular Velocity Formulas Explained
Formula 1: Displacement ÷ Time
Direct definition of angular velocity. Divide total angle swept (converted to radians) by time elapsed (in seconds).
Formula 2: Linear Velocity ÷ Radius
Relates linear and angular motion. A point on a rotating body at radius r moves at linear speed v = ω × r.
Formula 3: RPM Conversion
Converts RPM to rad/s. One revolution = 2π radians. Divide by 60 to convert per-minute to per-second.
Formula 4: Frequency
Converts frequency (Hz) to angular velocity. Each cycle is 2π radians, so multiply frequency by 2π.
Formula 5: Period Conversion
Converts period (time for one full rotation) to angular velocity. Period T is the reciprocal of frequency (T = 1/f).
Example Calculations
| Method | Input | Result (rad/s) | RPM |
|---|---|---|---|
| Displacement | 360° in 2 s | π ≈ 3.14 | 30 |
| Linear | v = 20 m/s, r = 5 m | 4.00 | 38.20 |
| RPM | 1200 RPM | 125.66 | 1200 |
| Frequency | 60 Hz | 376.99 | 3600 |
| Period | T = 0.5 s | 12.57 | 120 |
RPM to rad/s Conversion Reference
| RPM | rad/s | deg/s | rev/s |
|---|---|---|---|
| 1 | 0.1047 | 6.0 | 0.0167 |
| 60 | 6.2832 | 360 | 1.0 |
| 120 | 12.566 | 720 | 2.0 |
| 600 | 62.832 | 3,600 | 10.0 |
| 1200 | 125.66 | 7,200 | 20.0 |
| 3000 | 314.16 | 18,000 | 50.0 |
| 3600 | 376.99 | 21,600 | 60.0 |
Conversion factor: 1 RPM = π/30 rad/s ≈ 0.10472 rad/s
Real-World Applications
Electric Motors
Motor nameplate RPM is converted to rad/s for control system design and torque calculations.
Automotive Engineering
Engine crankshaft and wheel angular velocity are key to transmission and driveline design.
Robotics
Servo and stepper motor angular velocity determines joint speed and precision in robotic arms.
Physics Education
Angular velocity is a core concept in rotational kinematics for high school and university physics.
Centrifuges
Lab centrifuges operate at high RPM. Angular velocity determines centripetal force on samples.
AC Generators
Power grid generators spin at 3600 RPM (60 Hz) or 3000 RPM (50 Hz) to produce AC electricity.
Frequently Asked Questions
What is angular velocity?
Angular velocity (ω) is the rate of change of angular position of a rotating body. It tells you how fast something is spinning. The SI unit is rad/s. It differs from angular speed in that angular velocity is a vector quantity with a direction along the axis of rotation.
How do I convert RPM to rad/s?
Multiply RPM by π/30. For example, 1200 RPM × (π/30) = 125.66 rad/s. This is because one revolution = 2π radians, and one minute = 60 seconds, so the factor is 2π/60 = π/30 ≈ 0.10472.
What is the difference between angular velocity and angular frequency?
In many contexts they are the same (both equal ω = 2πf). Angular frequency is used in oscillation and wave physics, while angular velocity refers to the rotation of a rigid body. Both have units of rad/s.
How is angular velocity related to linear velocity?
v = ω × r, where v is linear velocity, ω is angular velocity, and r is the radius from the axis of rotation. A point farther from the center moves faster in a straight-line sense even though all points rotate at the same angular velocity.
What is the period of rotation?
The period T is the time for one complete rotation. T = 2π / ω = 1 / f. For a motor at 1200 RPM: T = 60/1200 = 0.05 seconds per revolution.
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