Angular Velocity
Free angular velocity calculator computes ω from angular displacement and time or linear velocity and radius. Interactive charts, step-by-step breakdowns, and shareable results for physics students and engineers.
About This Calculator
The Angular Velocity Calculator helps you compute the rotational speed of any object using one of two methods: from angular displacement and time (ω = Δθ/Δt) or from linear velocity and radius (ω = v/r). This tool is designed for physics students, engineers, and anyone studying rotational motion, circular dynamics, or mechanical systems.
Angular velocity (ω) measures how fast an object rotates or revolves relative to another point. The SI unit is radians per second (rad/s), but common conversions include RPM and degrees per second. The two supported formulas give identical results when applied consistently — choose the method that matches your available input values.
Formula Details
Method 1 — Angle & Time: ω = Δθ / Δt. Enter the change in angular position (in radians) and the time interval (in seconds). For example, rotating through π radians (180°) in 0.5 seconds gives ω = π / 0.5 ≈ 6.283 rad/s.
Method 2 — Velocity & Radius: ω = v / r. Enter the linear (tangential) velocity in m/s and the radius of rotation in meters. A point on a 0.5 m radius wheel moving at 15 m/s has ω = 15 / 0.5 = 30 rad/s.
The calculator also shows the relationship between inputs and output through interactive bar and pie charts, plus a detailed breakdown of all values used in the calculation.
Frequently Asked Questions
What is angular velocity?
Angular velocity (ω) measures how fast an object rotates or revolves around a center point. It is the rate of change of angular displacement over time, expressed in radians per second (rad/s). For example, a wheel spinning at 10 rad/s completes about 1.59 revolutions per second.
How do you calculate angular velocity from angle and time?
The basic formula is ω = Δθ / Δt where Δθ is the angular displacement in radians and Δt is the time in seconds. For example, if an object rotates through 6.28 rad (one full rotation) in 2 seconds, its angular velocity is 6.28 / 2 = 3.14 rad/s.
How do you calculate angular velocity from linear velocity and radius?
Use the formula ω = v / r, where v is the linear velocity in m/s and r is the radius of rotation in meters. For example, a point moving at 10 m/s on a circle of radius 2 m has an angular velocity of 10 / 2 = 5 rad/s.
What is the difference between angular velocity and linear velocity?
Linear velocity (v) measures straight-line distance traveled per unit time (m/s), while angular velocity (ω) measures the angle swept per unit time (rad/s). They are related by v = ω × r, where r is the radius. A point farther from the center has higher linear velocity at the same angular velocity.
How do you convert RPM to rad/s?
To convert RPM to rad/s, multiply by π / 30 (approximately 0.10472). For example, 3500 RPM × π / 30 = 366.52 rad/s. Conversely, to convert rad/s to RPM, multiply by 30 / π (approximately 9.5493).
What is the difference between angular velocity and angular frequency?
Angular frequency is the magnitude of angular velocity used for oscillatory motion like pendulums and springs. While angular velocity describes rotational motion as a vector, angular frequency is a scalar. Both use the same symbol ω and units (rad/s).
What is the angular velocity of the Earth?
The Earth's spin angular velocity is approximately 7.292 × 10⁻⁵ rad/s (about 1 revolution per 23.934 hours). Its orbital angular velocity around the Sun is about 1.991 × 10⁻⁷ rad/s. The linear velocity at the equator due to spin is about 465 m/s.
Does angular velocity apply to all rotating objects?
Yes, angular velocity applies to any rotating or revolving object including wheels, planets, gears, turbines, motors, and spinning tops. It is a fundamental concept in physics and engineering used to analyze rotational motion, centripetal force, rotational kinetic energy, and angular momentum.