Angular Velocity Calculator

Angular velocity measures how fast an object is rotating. It is the rate of change of the angle that an object sweeps per second around a certain center point. Both spinning wheels and orbiting planets, as well as records on turntables have angular velocities.

This calculator will calculate the angular velocity from four different values and shows the result in all common units for angular velocity. Enter either the angle and time, linear speed and radius, period or frequency.

What is angular velocity?

Angular velocity is represented by the Greek letter omega (ω) and describes a rotational motion. Normal speed indicates how far an object travels along a straight line per second. Angular velocity, on the other hand, indicates how much of an angle an object covers per second.

Angular velocity applies to two types of rotations. The first is rotation around an outside point, such as when a planet orbits the sun or a car turns a corner. The second is rotation around an object's own axis, such as Earth's daily spin or a basketball balanced on your fingertip.

The faster an object rotates, the greater its angular velocity is. Since angular velocity has a direction, it is technically a vector. However, in most everyday situations, knowing only its magnitude is sufficient and that's what this calculator provides.

Formula for angular velocity

There are mainly two ways to define angular velocity and this calculator uses both.

Calculation based on angle and time:

The first formula is similar to the linear velocity formula but uses angle instead of distance.

where Δθ is the amount of rotation and t is the time taken to make the rotation. When angles are measured in radians a unit of angular velocity is radians per second. Since one complete revolution is 2π radians, the angular velocity of a wheel turning at one revolution every two seconds (2π rad) would be π ≈ 3.14 rad/s.

Calculation based on linear speed and radius:

The second formula relates rotational motion to linear motion. A point on the rim of a rotating object moves in a circle, and its tangential speed v is determined by the angular velocity and radius.

Thus the angular velocity is equal to the linear velocity divided by the radius. The angular velocity of a point moving at 10 meters per second on a circle with a radius of 2 meters is 10/2 = 5 radians per second.

How to use this calculator:

Choose a method that fits your given values.

If you enter a value for the angular velocity and leave one of the other fields blank, the calculator will calculate ω. If you enter an known angular velocity and leave a field blank in any of the other fields, the calculator will recalculate the missing angle, time, radius, period or frequency.

Each field has its own unit conversion so you can enter angles in degrees or revolutions and radii in centimeters or inches. The angular velocity can be shown as rad/s, rev/min, rev/s or degrees per second, and all results are automatically converted.

Angular velocity units

Angular velocity can be expressed using a variety of units and the most common ones are listed in a row in the results area.

One unit of RPM is approximately 0.10472 radians per second; one revolution per second (1 rev/s or 1 hertz) is 2π radians per second, which is about 6.283 radians per second, or 60 RPM.

Conversion of RPM to rad/s

To convert a value from RPM to radians per second, multiply by 0.10472. For example, the angular velocity of an engine with a rotational speed of 3500 RPM is approximately 3500 × 0.10472 ≈ 366.5 rad/s. Conversely, you can divide a value in rad/s by 0.10472 or read the result directly from the results area in units of rev/min.

Angular velocity, frequency and period

The rate of rotation can also be expressed in terms of the frequency of repetition. The frequency f is the number of rotations per second and the period T is the time it takes for one rotation.

One revolution is equal to 2π radians so the product of frequency and 2π gives angular velocity. If a turntable rotates at 1 Hz (turns per second), then its angular velocity is 2π ≈ 6.28 rad/s, and its period is one second.

This is related to the angular frequency. The angular frequency and the angular velocity use the same symbol, the same formula ω = 2πf and the same unit rad/s. The difference is in the meaning. The angular frequency is a scalar used for any repetitive motion such as the oscillation of a pendulum. The angular velocity on the other hand is a vector quantity referring to actual rotation.

Example: Earth's rotation

The time it takes the Earth to rotate once relative to a distant star is about 23 hours and 56 minutes, or approximately 23.934 hours. Since one rotation corresponds to 2π radians, this gives an angular velocity of:

Even this extremely small angular velocity leads to a very high linear speed at the equator. The radius of the equator is about 6371 km. With the formula v = ωr, one gets approximately 6371000 × 7.29 × 10⁻⁵ ≈ 465 meters per second. This eastward acceleration is the reason why rockets are normally launched near the equator and in an easterly direction. The rocket thus receives a starting speed of almost half a kilometer per second for free, so to speak.

Tangential velocity and edge

In the rotation of a rigid body all points have the same angular velocity. However, points farther from the axis of rotation move faster along a straight line. This is because v = ωr increases as the radius becomes larger.

That's why the outer edge of a grinding wheel or record spins much faster than points closer to the center. And that's why a kid on the rim of a merry-go-round has to hold on tighter than one near the middle. A pair of belt drives uses the same principle: because the belt keeps speed the same at the rim, a smaller pulley must spin faster than a larger one.

Conservation of angular momentum

Angular velocity is a core component of an important conservation law. The angular momentum of a rotating object is proportional to its moment of inertia and its angular velocity, and this quantity remains constant in a closed system.

Figure skaters use this principle constantly: when they extend their arms, the moment of inertia increases and therefore rotation slows down; when they pull their arms in close to their body, the moment of inertia decreases so that the angular velocity must increase to keep the angular momentum constant, and the skater rotates faster. By the same physical law, collapsing stars accelerate and become rapidly rotating pulsars.

Applications of angular velocity

Engineers and scientists often work with angular velocity. Angular velocity determines the speed of motors, turbines, hard drive platters or turntables, as well as the cutting speed of drills or saws.

It also plays a role in the movement of planets, the swinging of pendulum clocks, the spinning of gyroscopes to stabilize ships or smartphones and the centripetal force required for car tires not to skid on curves. Anywhere an object rotates, angular velocity is a numerical value that indicates how fast it's rotating.

Common Mistakes

A common mistake is to confuse the units for angle. The formula ω = Δθ / t will give an angular velocity in radians per second only if θ is expressed in radians. If degrees or revolutions are used, then a conversion must be made. It is not correct to simply divide by the time and call it rad/s.

A common mistake is to confuse frequency and angular velocity. Since the two values differ by a factor of 2π, a frequency of 1 Hz corresponds to an angular velocity of 2π radians per second, not 1 radian per second. Also be careful that angular velocity is the result of dividing v by r, not multiplying them. Calculation tools can help avoid such mistakes.

This tool is used for general educational purposes and solving everyday problems. For technical work, safety measures or other risky activities the values must be checked according to the requirements of the particular project and data.