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
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
Angular Velocity
Rate of change of angular position
SI unit: rad/s
Linear Velocity
Speed at the edge of rotation
v = ωr
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.