velocity angular converter
Convert angular velocity (RPM, rad/s, deg/s, Hz, grad/s) with our Kinematics Engine. Features Motor-to-Wheel Linear Speed Bridge (v = ω·r), dynamic SVG gear train simulator, gearbox transmission ratio solver, and tachometer context matcher.
Unit ConvertersScientific Omni-Grid
Type into any rotational unit to calculate all others instantaneously.
Standard industrial & automotive shaft rotation
SI physical fundamental rotational velocity
Robotic joint & navigational angular rate
Rotational frequency cycles per second
Centesimal geodesic angular speed (400 grad = 2π)
= 20.0 ms per rotation
Dynamic Mechanical Visualizer
Simulated Rotational Velocity: 3000 RPM (0.12s/rev)
Motor-to-Wheel Linear Speed Bridge
Couples rotational shaft velocity with wheel radius to compute tangential road speed (v = ω · r).
How to Use the Angular Velocity Converter
Step-by-step guide to calculating rotational speeds, tangential linear velocities, and gear train ratios.
Enter Angular Speed in Any Unit
Input your known rotational value into any field (RPM, rad/s, °/s, Hz, or grad/s). All other rotational units synchronize instantly without latency.
Inspect Rotational Period & Frequency
Read the calculated cycle time per revolution (in seconds and milliseconds) and frequency in Hertz (rev/s) for dynamic cycle analysis.
Select Engineering Benchmarks
Click on preset benchmarks—such as Earth rotation, vinyl turntables, AC induction motors, or high-speed CNC spindles—for quick calibration.
Calculate Tangential Speed (v = ω · r)
Enter the wheel or pulley radius with units (cm, mm, m, in, ft) to determine tangential road speed in km/h, m/s, mph, and centripetal g-forces.
Simulate Gearbox Transmissions
Expand the transmission module to model reduction gears or overdrives using direct ratios or pinion-to-gear teeth counts (Z₁ : Z₂).
Copy & Export Kinematic Specs
Click "Copy Kinematic Spec" to export a complete formatted engineering summary including motor speed, radius, road velocity, and gear ratio.
Frequently Asked Questions
Essential technical answers regarding angular velocity, rotational mechanics, and transmission ratios.
Q:What is the difference between angular velocity and linear velocity?
Angular velocity (ω) measures how quickly an object rotates through an angle per unit of time (e.g., rad/s or RPM), independent of size. Linear velocity (v = ω · r) measures the actual tangential speed of a point at radius r along the circumference.
Q:How do you convert RPM to radians per second (rad/s)?
One full revolution is 2π radians and one minute is 60 seconds. Therefore, multiply RPM by 2π / 60 (approximately 0.10472). For example, 3,000 RPM × (π / 30) ≈ 314.16 rad/s.
Q:How do you convert radians per second (rad/s) to degrees per second (°/s)?
Since π radians equal 180 degrees, 1 rad/s equals 180 / π ≈ 57.2958°/s. To convert rad/s to degrees per second, multiply the value by approximately 57.2958.
Q:How does gear ratio affect output angular velocity and torque?
In an ideal transmission (neglecting friction), a speed reduction ratio (i > 1) divides output angular velocity by i while multiplying available output torque by i. Conversely, an overdrive (i < 1) increases rotational speed while proportionally reducing torque.
Q:What is centripetal acceleration and how does it scale with speed?
Centripetal acceleration (a꜀ = ω² · r) is the radial inward acceleration that keeps a rotating body on its circular path. Because it is proportional to the square of angular velocity (ω²), doubling rotational speed quadruples the centrifugal force experienced at the outer perimeter.
Q:What is the relationship between angular velocity and frequency (Hz)?
Frequency in Hertz represents complete cycles or revolutions per second (1 Hz = 1 rev/s = 60 RPM). Because one revolution contains 2π radians, angular velocity ω is related to frequency f by ω = 2π · f (or f = ω / 2π).
- ω (Omega): Angular Velocity (rad/s, rad/min).
- RPM: Rotational Speed (Revolutions Per Minute, rev/min).
- v: Tangential Linear Velocity (m/s, km/h).
- r: Radius of Wheel or Pitch Circle (m, cm, mm).
- ac: Centripetal Acceleration (m/s², g).
- i: Transmission Gear Ratio (Z2 / Z1).
- Z1, Z2: Tooth Count of Driver Pinion and Driven Gear.
Velocity angular converter Table
| Unit | rad/s | deg/s | rev/s | rev/min (RPM) |
|---|---|---|---|---|
| 1 rad/s | 1 | 57.2957795131 | 0.1591549431 | 9.5492965855 |
| 1 deg/s | 0.0174532925 | 1 | 0.0027777778 | 0.1666666667 |
| 1 rev/s | 6.2831853072 | 360 | 1 | 60 |
| 1 rev/min (RPM) | 0.1047197551 | 6 | 0.0166666667 | 1 |
What is velocity angular converter?
Convert angular velocity across RPM, rad/s, deg/s, Hz, and grad/s effortlessly with our advanced Kinematics Engine. This platform is engineered for mechanical design and motion control, featuring a dynamic SVG gear train simulator to visualize torque and speed transfer. The integrated gearbox transmission ratio solver takes the guesswork out of mechanical reduction. Crucially, the Motor-to-Wheel Linear Speed Bridge utilizes the formula v = ω · r to instantly translate rotational motor output into tangible linear vehicle speed.
Practical Calculation Example
An automotive engineer in Detroit is tuning a new electric vehicle powertrain. They input the motor's peak output of 12,000 RPM into the Kinematics Engine. Using the Motor-to-Wheel Linear Speed Bridge with a known tire radius of 0.35 meters, they instantly calculate the exact linear highway speed in miles per hour. The gear train simulator helps them visually balance the planetary reduction ratio, and they export the finalized kinematic parameters directly to their suspension simulation software.
Linear vs. Angular Momentum: The Physical Bridge
Rotational dynamics is the exact rotational analog of classical translational Newtonian mechanics:
Gear Train & Transmission Kinematics
In physical powertrains, motors rarely couple directly to final drives. A gearbox provides mechanical advantage governed by the gear ratio i:
Neglecting frictional losses, mechanical power is conserved: P = τin · ωin = τout · ωout. Consequently, the output driving torque is:
where η is mechanical efficiency. A 3:1 speed reducer reduces rotational speed to one-third while tripling available driving torque.
Step-by-Step Kinematics Engineering Tutorials
Tutorial 1: Mobile Robot Chassis Sizing
Scenario: A mobile robotic platform utilizes wheels with radius r = 12 cm (0.12 m). The brushless hub motor operates at 1,200 RPM. Calculate the robot's linear travel speed in m/s and km/h.
- Convert RPM to radians per second:
ω = 1,200 × (2π / 60) ≈ 125.664 rad/s - Compute tangential linear velocity:
v = ω · r = 125.664 rad/s × 0.12 m ≈ 15.08 m/s - Convert to km/h:
vkm/h = 15.08 × 3.6 ≈ 54.29 km/h
Tutorial 2: CNC Spindle Surface Speed
Scenario: An end mill with diameter D = 10 mm (radius r = 5 mm = 0.005 m) rotates at 18,000 RPM. Find the cutting tip peripheral velocity:
- Angular velocity:
ω = 18,000 × (2π / 60) ≈ 1,884.96 rad/s - Peripheral tip speed:
v = 1,884.96 rad/s × 0.005 m = 9.42 m/s = 565.5 m/min
History and Origin
The formalized study of angular velocity became critical during the Industrial Revolution with the rapid advancement of the steam engine. James Watt's invention of the centrifugal governor in 1788 relied on rotational speed to mechanically regulate steam flow, laying the absolute groundwork for modern control theory. Today, the radian per second is the SI unit of angular velocity, providing a mathematically elegant way to link rotational physics with linear dynamics.