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Angular Velocity Calculator

Choose the method from the list and fetch the required values. This online calculator will readily calculate the angular velocity using the selected method, with the steps shown instantly.

Choose the method:

To Calculate:

Angular Velocity Angle Change Time

To Calculate:

Angular Velocity Linear Velocity Radius

Angle Change (Δα):

deg ▾

Time (t):

sec ▾

Angular Velocity (ω):

rad/s ▾

Linear Velocity (v):

m/s ▾

Radius (r):

m ▾

RPM:

Radius:

m ▾

Related

Our online angular velocity calculator allows you to determine the angular velocity of an object moving along a circular path. It supports three main methods for calculation:

Using angular displacement (angle difference)
Using radial (linear) velocity
Converting between angular velocity and linear velocity

Below, we explain the formulas, provide step-by-step examples, and show how to use the calculator. You can also try our online velocity calculator for general speed and velocity calculations.

What is Angular Velocity?

Angular velocity measures how quickly an object rotates around a reference point. It is the rate of change of an object’s angular position over time. Symbolically, it is represented by \(ω\) and commonly measured in radians per second (rad/s).

Example:

Earth rotates once around its axis every 24 hours (360°). Its angular velocity is:

\(ω = \frac{360°}{24 h} \) or \(ω = \frac{2\pi rad}{86400 s}\)

Formulas for Angular Velocity

The formula you use depends on the information available.

1. Using Angle Difference:

When the change in angle and time are known:

\( \omega = \frac{\alpha_2 - \alpha_1}{t} = \frac{\Delta \alpha}{t} \)

Where:

\(\alpha_1, \alpha_2\) = initial and final angles
\(\Delta\alpha\) = change in angle
t = time interval

You can also use our acceleration calculator to find the rate of change of velocity.

2. Using Radial (Linear) Velocity:

When linear velocity and radius are known:

\(v = \omega \times r \quad \Rightarrow \quad \omega = \frac{v}{r}\)

Where:

r = radius
v = linear velocity

3. Converting Angular Velocity and Linear Velocity:

To convert between linear velocity and angular velocity using RPM:

\(v = r \times RPM \times 0.10472\)

Where:

v = linear velocity in m/s
r = radius in meters
RPM = angular velocity in revolutions per minute

Units of Angular Velocity

Angular velocity can be expressed in:

Radians per second (rad/s)
Revolutions per minute (RPM)
Hertz (Hz) – number of rotations per second

Conversion formulas used by the calculator:

\(1 RPM = 0.10472 rad/s = 0.01667 Hz\)

\(1 Hz = 6.283 rad/s = 60 RPM\)

Step-by-Step Calculations

Example 1: Using Angle Difference

An object moves from 5 rad to 25 rad in 16 seconds. Find angular velocity.

Solution:

\(\Delta\alpha = 25 - 5 = 20 rad\)

\(\omega = \frac{20}{16} = 1.25 rad/s\)

Example 2: Using Linear Velocity

A Ferris wheel has radius 10 m and linear velocity 35 m/s. Find angular velocity.

Solution:

\(\omega = \frac{v}{r} = \frac{35}{10} = 3.5 rad/s\)

Example 3: Converting RPM to Linear Velocity

Object radius = 1.5 m, RPM = 10. Find linear velocity.

Solution:

\(v = r \times RPM \times 0.10472 = 1.5 \times 10 \times 0.10472 = 1.57 m/s\)

Using the Angular Velocity Calculator

Our calculator provides accurate results using the methods above. Simply:

Select the method (angle difference, radial velocity, or RPM conversion)
Enter the required values
Click Calculate

Outputs include:

Angular velocity (rad/s, degrees/s, RPM, Hz)
Linear velocity (m/s, km/h, miles/h)
Change in angle
Time interval
Radius in multiple units (m, cm, ft, in, yd)

FAQs

Is angular velocity the same everywhere on a rotating object?

Yes, angular velocity is constant along the axis of rotation. Tangential velocity varies with distance from the axis.

How can you find velocity without time?

Use the distance traveled in place of time in the relevant equations. Convert units as needed, e.g., multiply ft/s by 0.68 to get mph.

What is Omega (ω)?

Angular velocity, denoted \(\omega\), measures angular displacement per unit time. Its value in rad/s is related to frequency by:

\(\omega = 2\pi f\), so 1 Hz = 6.28 rad/s

Why is v = ωR?

This formula converts angular velocity into linear velocity for a point on a rotating object. If the rotation period is t, \(\omega = \frac{2 \pi}{t}\).

Conclusion

Angular velocity is a fundamental parameter for circular motion, helping determine rotation rates and link linear and rotational motion. Our calculator simplifies these calculations for accurate results in multiple units.

References

Wikipedia: Angular velocity. Lumen Learning: Angular velocity units. HyperPhysics: Rotational quantities.

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