Angular Velocity Calculator

Calculate angular velocity using multiple methods with precise physics formulas and conversions

Calculate Angular Velocity

Total angle rotated

Duration of rotation

Angular Velocity Results

0.0000
rad/s
0.0000
rad/s (SI unit)

Formula used: ω = Δα / Δt

Calculation: ω = 0.0000 rad / 0.0000 s = 0.0000 rad/s

Example Calculations

Earth's Rotation

Problem: Earth rotates 2π radians in 23.934 hours

Given: Δα = 2π rad, Δt = 23.934 h = 86,162.4 s

Formula: ω = Δα / Δt

Solution: ω = 2π / 86,162.4 = 7.292 × 10⁻⁵ rad/s

Car Tire Rotation

Problem: Car moving at 60 km/h with tire radius 0.3 m

Given: v = 60 km/h = 16.67 m/s, r = 0.3 m

Formula: ω = v / r

Solution: ω = 16.67 / 0.3 = 55.56 rad/s = 530.5 RPM

Engine RPM to Angular Velocity

Problem: Engine running at 3000 RPM

Given: f = 3000 RPM = 50 Hz

Formula: ω = 2πf

Solution: ω = 2π × 50 = 314.16 rad/s

Key Concepts

ω

Angular Velocity

Rate of change of angular position

SI unit: rad/s

v

Linear Velocity

Speed at the edge of rotation

v = ωr

f

Frequency

Number of rotations per second

Unit: Hz

Unit Conversions

1 RPM equals:

  • • 0.10472 rad/s
  • • 0.01667 Hz
  • • 6 deg/s

1 Hz equals:

  • • 6.283 rad/s
  • • 60 RPM
  • • 360 deg/s

1 rad/s equals:

  • • 9.549 RPM
  • • 0.1592 Hz
  • • 57.296 deg/s

Key Formulas

Angular Displacement

ω = Δα / Δt

Linear Velocity

ω = v / r

Frequency

ω = 2πf

Period

ω = 2π / T

Understanding Angular Velocity

What is Angular Velocity?

Angular velocity describes how fast an object rotates around an axis. It measures the rate of change of angular position with respect to time. Unlike linear velocity, which describes motion along a straight path, angular velocity describes rotational motion.

Types of Angular Motion

  • Orbital Motion: Movement of center of mass around an external point (planets around Sun)
  • Spin Motion: Rotation around object's own center of mass (Earth's rotation)

Key Relationships

  • Linear Velocity: v = ωr (velocity increases with radius)
  • Period: T = 2π/ω (time for one complete rotation)
  • Frequency: f = ω/(2π) (rotations per second)
  • Angular Acceleration: α = dω/dt (rate of change of angular velocity)

Vector Nature

Angular velocity is a pseudovector. Its direction is perpendicular to the plane of rotation, following the right-hand rule: curl fingers in direction of rotation, thumb points in direction of angular velocity vector.

Real-World Applications

🚗 Automotive

Engine RPM, wheel rotation, turbocharger speed, and transmission gear ratios all involve angular velocity calculations.

🌍 Astronomy

Planetary rotation, orbital motion, pulsar spin rates, and galaxy rotation curves are described using angular velocity.

⚙️ Machinery

Industrial motors, centrifuges, washing machines, and turbines operate at specific angular velocities for optimal performance.

🏃‍♂️ Sports

Figure skating spins, gymnastics rotations, ball spin in various sports, and bicycle wheel rotation all involve angular velocity.

🎮 Electronics

Hard drive spindles, cooling fans, CD/DVD players, and gyroscopes in smartphones use angular velocity measurements.

🏭 Manufacturing

Lathes, mills, conveyors, and robotic arms require precise angular velocity control for accurate manufacturing processes.