Angular Velocity Calculator

The Angular Velocity Calculator converts any rotation rate between RPM, rad/s, deg/s, Hz, revolutions per second, period (s per revolution), and revolutions per hour or day, all in a single form. It also computes angular velocity from a linear speed and a radius using ω = v/r, and adds the tangential speed and centripetal acceleration at any radius you supply. A live SVG disk spins at exactly the computed angular velocity so you can see — not just read — what the number means, and a comparison panel shows where your ω falls among real-world rotations (clock hands, Earth, vinyl records, hard drives, washing machines, jet engines, ultracentrifuges).

How to Use This Angular Velocity Calculator

1. Choose the mode: Convert an angular value if you already have ω in some unit and want to express it in others; From linear motion (v, r) if you know a tangential speed and radius and want ω.
2. In convert mode, enter the value and pick any of the nine angular units. Optionally enter a radius and the calculator will also report the tangential speed and centripetal acceleration at that radius.
3. In linear motion mode, enter the linear speed v and radius r. The calculator applies ω = v / r and gives you ω together with the full conversion table.
4. Press Calculate to see ω in every unit, the matching real-world rotation, the spinning-disk animation, and a step-by-step derivation.

What Makes This Calculator Different

Angular Velocity Formula

The angular velocity \(\omega\) of a rotating object is the rate at which its angular position changes:

\[ \omega \;=\; \dfrac{d\theta}{dt} \quad\text{(rad/s)} \]

For uniform rotation, this is the same as the total angle swept (in radians) divided by the time taken. The most common conversions between everyday units are:

\[ \omega_{\text{rad/s}} \;=\; \text{RPM} \cdot \dfrac{2\pi}{60} \;=\; f \cdot 2\pi \;=\; \dfrac{2\pi}{T} \]

where \(f\) is the rotation frequency in hertz and \(T\) is the period in seconds. The reverse direction is

\[ \text{RPM} \;=\; \omega_{\text{rad/s}} \cdot \dfrac{60}{2\pi}, \quad f \;=\; \dfrac{\omega}{2\pi}, \quad T \;=\; \dfrac{2\pi}{\omega} \]

Relating Angular and Linear Motion

If an object is moving on a circular path of radius \(r\), its linear (tangential) speed \(v\) and angular velocity \(\omega\) are related by

\[ v \;=\; \omega \, r \quad\Longleftrightarrow\quad \omega \;=\; \dfrac{v}{r} \]

Likewise, the centripetal acceleration that always points to the center of rotation is

\[ a \;=\; \omega^{2} \, r \;=\; \dfrac{v^{2}}{r} \]

The calculator computes both \(v\) and \(a\) automatically whenever you provide a radius, and also expresses \(a\) in multiples of standard gravity (g) so you can compare to real-world experience.

Worked Example: Washing Machine Spin Cycle

A washing-machine drum of radius 25 cm spins at 1200 RPM during the spin cycle.

• Convert to rad/s: \(\omega = 1200 \cdot 2\pi / 60 \approx 125.66\) rad/s.
• Tangential speed at the drum wall: \(v = \omega r = 125.66 \cdot 0.25 \approx 31.4\) m/s ≈ 113 km/h — that's why wet clothes hit the drum hard.
• Centripetal acceleration: \(a = \omega^{2} r \approx 3948\) m/s² ≈ 402 g — about 400 times Earth's gravity, which is what flings the water through the drum holes.

Worked Example: Car Wheel at 100 km/h

A car wheel of outer radius 31 cm (typical passenger tire) rolls without slipping at a vehicle speed of 100 km/h ≈ 27.78 m/s.

• Angular velocity: \(\omega = v / r = 27.78 / 0.31 \approx 89.6\) rad/s.
• In RPM: \(\omega \cdot 60 / (2\pi) \approx 856\) RPM — comfortably below the typical 7000 RPM redline of a passenger-car engine, which is why automotive transmissions exist.
• Period: \(T = 2\pi/\omega \approx 0.070\) s per revolution.

Real-World Angular Velocity Reference

Hz vs RPM vs rad/s — Which Do I Use?

All three describe exactly the same physical rotation, just with different unit conventions:

• RPM (revolutions per minute) is the everyday engineering unit for motors, fans, drives, and turntables.
• Hz (revolutions per second) is the same as rev/s and matches the rotation frequency. 1 Hz = 60 RPM.
• rad/s (radians per second) is the SI unit, used in physics formulas because the math is cleaner: \(v = \omega r\), \(a = \omega^{2} r\), \(\theta = \omega t\). One full revolution equals \(2\pi\) radians, so 1 Hz = 2π rad/s ≈ 6.283 rad/s, and 1 RPM = 2π/60 rad/s ≈ 0.1047 rad/s.

Why Use Radians Instead of Degrees?

A radian is defined as the angle subtended by an arc whose length equals the radius, which makes \(s = r\theta\) and \(v = r\omega\) hold without any conversion factor. Using degrees would introduce a clumsy \(\pi/180\) factor into every derivation. That is why physics and engineering formulas universally use rad/s, even though everyday measurements (RPM, degrees per second) feel more intuitive. The calculator does the conversion in both directions so you can stay in whichever unit makes the problem clearest.

Frequently Asked Questions

How do I convert RPM to rad/s?
Multiply RPM by 2π/60. So 60 RPM equals 60 × 2π / 60 = 2π ≈ 6.283 rad/s. The calculator does this automatically when you change the input unit.

How do I convert rad/s to RPM?
Multiply rad/s by 60/(2π). So 10 rad/s equals 10 × 60 / (2π) ≈ 95.49 RPM. The full conversion table in the result panel shows every common unit at once.

How is angular velocity related to linear speed?
For a point moving on a circle of radius r, the tangential speed v equals the product ω·r. Therefore ω = v/r. The linear-motion mode applies this directly.

What is the difference between Hz and rad/s?
Hertz counts complete revolutions per second; rad/s measures the angular sweep in radians per second. One revolution = 2π radians, so 1 Hz = 2π rad/s. The two describe the same rotation in different unit systems.

How do I find the period of a rotation?
The period T is the time for one complete revolution, equal to 2π/ω in seconds, or 60/RPM in seconds. At 6000 RPM the period is 60/6000 = 0.01 s = 10 ms per revolution.

What is centripetal acceleration?
For circular motion at angular velocity ω and radius r, the centripetal acceleration is ω²r, always pointing toward the center. The calculator reports this whenever a radius is supplied and also expresses it in multiples of standard gravity for easy intuition.

Can I use this calculator on a phone?
Yes. The layout switches to a single-column stack on screens narrower than 900 px, keeping every input, preview, and result panel readable without horizontal scrolling.