Last updated: July 16, 2026
Angular Velocity Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
Angular velocity (ω) quantifies rotational speed as the rate of angular displacement change in radians per second. This calculator computes ω via three physics formulas: ω = Δθ/Δt (angle over time), ω = v/r (linear velocity over radius), and ω = 2πf (from frequency). It also derives period, frequency, and RPM from the result. Applications span rotating machinery, AC motors, planetary motion, and sports physics.
Angular velocity is calculated using omega equals delta-theta divided by delta-t for angle and time, omega equals v divided by r for linear velocity and radius, or omega equals 2-pi times f for frequency. The result is in radians per second, with conversions to RPM, degrees per second, and more.
Key Takeaways
- Angular velocity (ω) is the rate of change of angular displacement, measured in rad/s.
- Three equivalent formulas: ω = Δθ/Δt, ω = v/r, and ω = 2πf — choose based on available data.
- Angular velocity is a pseudovector; its direction follows the right-hand rule along the rotation axis.
- Linear and angular quantities are linked: v = ωr, a_c = ω²r, KE = ½Iω².
- Common conversions: 1 RPM = 2π/60 ≈ 0.1047 rad/s; 1 rad/s ≈ 9.549 RPM.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
ω = Δθ / Δt
Where:
- ω=Angular Velocity(rad/s)
- Δθ=Angle Change(rad)
- Δt=Time Interval(s)
- v=Linear Velocity(m/s)
- r=Radius(m)
- f=Frequency(Hz)
Worked Examples
Earth's Rotation
Calculate the angular velocity of Earth's daily rotation: 360 degrees in 24 hours.
- 1Convert angle: 360° × (π/180) = 2π ≈ 6.2832 radians
- 2Convert time: 24 hours × 3600 s/h = 86400 seconds
- 3Apply formula: ω = Δθ / Δt = 6.2832 / 86400
- 4Result: ω ≈ 7.272 × 10⁻⁵ rad/s
Car Wheel
A car travelling at 10 m/s has wheels with radius 0.35 m. Find the angular velocity.
- 1Use formula: ω = v / r
- 2Substitute: ω = 10 m/s / 0.35 m
- 3Result: ω ≈ 28.57 rad/s
- 4Equivalent to: 28.57 × 60/(2π) ≈ 272.8 rpm
Electric Motor at 50 Hz
An electric motor operates at 50 Hz mains frequency. Calculate its angular velocity.
- 1Use formula: ω = 2π × f
- 2Substitute: ω = 2π × 50 Hz
- 3Result: ω = 314.16 rad/s
- 4Equivalent to: 314.16 × 60/(2π) = 3000 rpm
Introduction
Angular velocity (ω, omega) is a fundamental quantity in rotational mechanics that describes how quickly an object rotates about a fixed axis. Measured in radians per second (rad/s), it represents the rate of change of angular displacement over time. Unlike linear velocity which describes straight-line motion, angular velocity characterizes circular and rotational motion — from the spin of a planet to the rotation of a motor shaft. This calculator supports three methods: from angle change and time (ω = Δθ/Δt), from linear velocity and radius (ω = v/r), and from rotational frequency (ω = 2πf).
Definition and Core Formula
Angular velocity is defined as the rate of change of angular displacement with respect to time. The fundamental formula is ω = Δθ / Δt, where Δθ is the angle swept (in radians) and Δt is the elapsed time (in seconds). The SI unit of angular velocity is radians per second (rad/s). For a full rotation, θ = 2π radians, so a body completing one revolution per second has ω = 2π rad/s ≈ 6.283 rad/s. Angular velocity is a vector quantity — its direction is given by the right-hand rule along the axis of rotation. For scalar problems (2D rotation), only the magnitude is used. See also angular acceleration for how ω changes over time.
Three Calculation Methods
Method 1 — Angle and Time (ω = Δθ/Δt): The most direct method. Measure how many radians the object sweeps in a given time interval. Useful when you can observe angular displacement directly, as with a rotating disk or pendulum. Method 2 — Linear Velocity and Radius (ω = v/r): When you know the speed of a point on the rim and the distance from the axis, ω equals the tangential speed divided by the radius. This is practical for wheels and gears where the rim speed is measurable. Method 3 — Frequency (ω = 2πf): Rotational frequency f (in Hz, meaning rotations per second) is related to angular velocity by the factor 2π. This form is widely used in electrical engineering and AC motor analysis. RPM is first converted to Hz by dividing by 60: f = n/60. For authoritative definitions, see the NIST guide to SI units.
Common Unit Conversions
Angular velocity can be expressed in several units depending on context: - rad/s (SI standard): base unit, used in physics equations - RPM (revolutions per minute): common in engineering, n = ω × 60/(2π) - deg/s (degrees per second): used in navigation and robotics, °/s = ω × 180/π - rev/s (revolutions per second): ω = 2πf, so rev/s = ω/(2π) For practical conversions: 1 rad/s ≈ 9.549 RPM, 1 RPM ≈ 0.1047 rad/s, 1 rev/s = 2π rad/s ≈ 6.283 rad/s. The angular momentum calculator uses ω directly as L = Iω.
Real-World Applications
Angular velocity appears in a vast range of engineering and scientific contexts: Mechanical Engineering: Gear trains, turbines, flywheels, and centrifuges are all characterized by their angular velocity. Knowing ω allows engineers to calculate bearing loads, power transmission efficiency, and centripetal stresses. Electrical Engineering: AC generators and motors operate at angular frequencies determined by the mains frequency (50 or 60 Hz). ω = 2πf = 314.16 rad/s at 50 Hz; synchronous motor speeds are ω/pole pairs. Astronomy: Planetary rotation rates are described in rad/s. Earth's sidereal angular velocity is ≈ 7.292 × 10⁻⁵ rad/s. Sports Science: The angular velocity of a thrown ball or golf club determines spin rate and thus aerodynamic forces. Gyroscopes in phones and drones measure angular velocity via MEMS sensors. For further reading, see NASA Glenn Research Center — Rotational Motion.
Vector Nature and Sign Convention
Angular velocity is properly a pseudovector (axial vector) whose direction is defined by the right-hand rule: curl the fingers of the right hand in the direction of rotation, and the thumb points in the direction of ω. For counterclockwise rotation when viewed from above, ω is positive (pointing upward); clockwise rotation gives a negative ω (pointing downward). In scalar 2D problems this convention means: - Counterclockwise (CCW) → ω > 0 - Clockwise (CW) → ω < 0 This matters in mechanics problems involving angular momentum conservation, torque equations (τ = dL/dt), and gyroscopic effects. Many calculators and textbooks work only with magnitudes, but always track the sign when solving dynamics problems. The angular displacement calculator uses consistent sign conventions with this approach.
Quick Reference Card
Angular Velocity Quick Reference
Quick reference • Angular Velocity Calculator
ω = Δθ/Δt = v/r = 2πfValid range: ω > 0 for CCW rotation; ω < 0 for CW; ω = 0 for no rotation
Common Values
⚠ Watch Out
- •Always convert angles to radians before applying ω = Δθ/Δt — degrees will give incorrect results.
- •The formula ω = v/r requires consistent units: v in m/s and r in metres gives ω in rad/s.
- •RPM is not the same as Hz; RPM must be divided by 60 to get Hz before using ω = 2πf.
- •Angular velocity is a vector — in 3D problems, track the direction using the right-hand rule.
- •Do not confuse angular velocity (ω, rad/s) with angular frequency (also ω) in wave physics — context matters.
Pro Tips
- →Use ω = 2πf when working with AC circuits or motors — the supply frequency f is usually given directly.
- →Use ω = v/r for wheels and pulleys where the rim speed is easier to measure than the angle.
- →Period and angular velocity are reciprocals: T = 2π/ω. Measuring T is often easier than measuring ω directly.
- →For unit conversions, remember: multiply rad/s by 60/(2π) ≈ 9.549 to get RPM; divide RPM by 9.549 to get rad/s.
FAQs
What is angular velocity and how does it differ from linear velocity?
Angular velocity (ω) measures how fast an object rotates — specifically, the rate of change of its angular displacement. Linear velocity (v) measures how fast a point moves in a straight line. They are related by v = ω × r, where r is the distance from the rotation axis. A point on the edge of a wheel moves faster (higher v) than a point near the center, but both share the same ω.
What are the SI units of angular velocity?
The SI unit is radians per second (rad/s). Since radians are dimensionless (they are the ratio of arc length to radius, both in metres), angular velocity effectively has units of s⁻¹. Other common units are RPM (revolutions per minute), degrees per second (°/s), and Hz (when treating ω/(2π)).
How do I convert RPM to rad/s?
Multiply by 2π and divide by 60: ω (rad/s) = n (RPM) × 2π / 60 ≈ n × 0.10472. For example, 1500 RPM = 1500 × 0.10472 = 157.08 rad/s. Conversely, 1 rad/s = 60/(2π) ≈ 9.549 RPM.
What is the relationship between angular velocity and frequency?
Angular velocity and frequency are related by ω = 2πf, where f is the frequency in Hz (cycles per second). This means ω is the angular frequency — it represents the same oscillation rate but measured in radians rather than cycles. For a 50 Hz motor, ω = 2π × 50 = 314.16 rad/s.
Can angular velocity be negative?
Yes. Angular velocity is a vector quantity (technically a pseudovector). The sign indicates the direction of rotation: positive values typically represent counterclockwise rotation (when viewed from a defined positive axis direction), while negative values represent clockwise rotation. In practice, many engineering calculations use only the magnitude.
What is the difference between angular velocity and angular speed?
Angular speed is the magnitude of angular velocity — it is always non-negative and describes how fast the rotation is, without indicating direction. Angular velocity includes both magnitude and direction (given by the rotation axis vector via the right-hand rule). For most scalar calculations, the two terms are used interchangeably, but in vector mechanics the distinction matters.