Angular Velocity Calculator (2023)

This angular velocity calculator is an easy-to-use tool that instantly answers questions such as:"How to find angular velocity?". you will find in the textDifferent Angular Velocity Formulas, learn about the different units of angular velocity, and finally estimate the angular velocity of the Earth!

have you ever thought aboutWhat is the relationship between angular velocity and angular frequencyyes? Or where does angular velocity apply? Read on to learn and become a circular motion expert.

πŸ™‹Visit ourCircular Motion CalculatorLearn about other important physical quantities of bodies of revolution.

What is angular velocity?

Angular velocity describes the rotational motion of an object. It measures how fast they move around a given center of rotation. We can imagine two different types of rotations. the first describedThe motion of the center of mass of an object around a point in space, which we can describe as the origin. Some examples include planets orbiting the sun or cars pulling off the highway.

another is ringingThe body rotates around its own center of mass–turnover(Not to be confused with the quantum property of particles, also known as spin). You've probably seen basketball players spin the ball with their fingers.

In general, we can say that the faster the motion, the greater the angular velocity. We need to continue using the angular velocity equation described in the next section to define some specific values.

Angular Velocity Formula

Angular Velocity Calculator (1)

This angular velocity calculator uses two different angular velocity formulas depending on your input parameters.

The first angular velocity equation is similar to the linear velocity equation:

oh=A2-A1time=DAtime,\omega = \frac{\alpha_2 - \alpha_1}{t} = \frac{\Delta\alpha}{t},oh=timeA2​-A1​​=timeDA​,

WhereA1\alfa_1A1​UA2\alfa_2A2​are the two values ​​of the angle on the circle, iDA\Delta\alphaDAis their difference.timetimetimeis the time at which the angle change occurs. As you can see, for normal velocity there is a ratio of position displacement to period, and here we use angles instead of distances.

Angular Velocity Calculator (2)

According to the cross-product relationship between linear velocity and radius, another formula for angular velocity can be derived, namely:

v⃗=oh⃗×right⃗\vec{v} = \vec{\omega} \times \vec{r}v=oh×right

We can rewrite this expression to get the angular velocity equation:

ohβƒ—=rightβƒ—Γ—vβƒ—βˆ£right∣2,\vec{\omega} = \frac{\vec{r} \times \vec{v}}{{|r|}^2},oh=∣right∣2rightΓ—v​,

WhereAll these variables are vectors,you∣right∣|r|∣right∣Indicates the absolute value of the radius. In fact, angular velocity is a pseudo-vector whose direction is perpendicular to the plane of rotational motion.

angular velocity unit

There are many units of angular velocity, the units we use in the angular velocity calculator are as follows:

  • rad/sfromradial per second– Define the first formula directly from the angular velocity. It tells us how much rotation (or angle) the body has moved in a certain amount of time.

  • revolutions per minutefromrevolutions per minute– The most common unit in practice. This allows you to describe how fast a wheel or motor is turning. You can easily imagine the difference between10U100 t/min。

  • hertzfromheck– Same unit as used for frequency, but rarely used for angular velocity. is somewhat similarrevolutions per minute, which tells us how many full rotations we have made in a certain amount of time. The difference is that the basic unit of time in the past was minutes, but here it is seconds.

Of course, all these units of angular velocity can be converted to each other using the following relationship:

1 RMP = 0,10472 rad/s = 0,01667 Hz,

or vice versa:

1 hertz = 6,283 radians per second = 60 tpm。

Angular velocity and angular frequency

See the definition of angular frequency:

oh=2PIF,Ξ© = 2pi f,oh=2PIF,

WhereFFFis the frequency. As we can see, it is represented by the same letters. Also, the unit of angular frequency israd/s, which is exactly the same as the angular velocity. Therefore, it is possible to ask: "What is the difference between angular velocity and angular frequency?".

The answer is relatively simple.The relationship between angular frequency and angular velocity is similar to the relationship between velocity and velocity. The first is the magnitude of the other; in other words, angular frequency is a scalar, while angular velocity is a (pseudo)vector.

When we talk about angular frequency, we often use angular frequency.harmony movement, a simple pendulum is an example. As you can imagine, motion doesn't have to be represented by a standard rotation, but simply by a motion that periodically repeats its position. However, angular velocity is strictly related to motion around a point. That's why we can sayThe angular frequency is a more general quantity, which we can use to describe a wide variety of bodily problems. In contrast, angular velocity involves only rotational motion.

πŸ™‹ We have a special tool that explains how to calculate the angular frequency. be sure to check it outAngular Frequency Calculator!

How did you find the angular velocity of the earth?

What if we use an angular velocity calculator? Let's estimate the angular velocity of the Earth! First, we consider the rotation speed. we knowEarth rotates once relative to a distant star,About23:56:4,That's it23.934 SATA. The whole rotation is an angle2Ο€ work, so the resulting angular velocity is:

oh1=2PIrightAD/23.934H=0,2625rightAD/H=0,00007292rightAD/S,\begin{split}\omega_1 &= 2\pi\ \rm rad / 23.934\ h\\&= 0.2625\ \rm rad/h\\&= 0.00007292\ \rm rad/s,\end{division}oh1​​=2PIRadian/23.934H=0,2625Work=0,00007292rad/s,​

from7.292Γ—10-5rightAD/S7.292 \times 10^{-5}\ \rm radians/second7.292Γ—10-5rad/s(using scientific notation).

Now that we know the angular velocity of the Earth's rotation, we can estimate the linear velocity at the equator. For this, we need a radius of approximately the Earth6 371 kilometers. All we have to do is plug these values ​​into the second angular velocity formula:

v1=right1oh1=6371kMediumβ‹…7.292⁣⁣×⁣⁣10-5rightAD/S=0,4646kMedium/S=464,6Medium/S\small\begin{split}v_1 &= r_1 \omega_1\\&= 6371\ \rm km \cdot 7,292\!\times\!10^{-5}\ rad/s\\&= 0,4646\ \ rm km /s\\&= 464.6\ \rm m/s\end{splitsen}v1​​=right1​oh1​=6371kilometerβ‹…7.292Γ—10-5rad/s=0,4646km/s=464,6Mrs​

All you have to do is calculate the linear velocity relative to the center of the earthMultiply this result by the cosine of your city's latitude。

By the way, have you ever wondered why rockets are usually launched from spaceports near the equator and not from the poles? well, almost one500 m/sInitial boost is an important part of terminal speed. Putting the launch point as close to the equator as possible reduces the amount of fuel needed to accelerate the rocket.

Angular Velocity Calculator (3)

Then we can ask again how to find the Earth's angular velocity, but this time it's orbital velocity. All calculations are simulated, but we need to change the time23.943 SATAUp to a year, that's all365,25dawn. The angle change is the same, a full circle.

oh2=2PIrightAD/23.934H=0,0000001991rightAD/S=1.991⁣⁣×⁣⁣10-7rightAD/S,\begin{split}\omega_2 &= 2\pi\ \rm rad / 23,934\ h\\&= 0,0000001991\ \rm rad/s\\&= 1,991\!\times\!10^{-7} \\ rm rad/s,\end{split}oh2​​=2PIRadian/23.934H=0,0000001991rad/s=1.991Γ—10-7rad/s,​

and the Earth's linear velocity relative to the Sun (for the mean radius1.496⁣⁣×⁣⁣108kMedium\klein 1.496\!\times\!10^8\ \rm km1.496Γ—108kilometer) yes:

v2=1.496Γ—108kMediumβ‹…1.991Γ—10-7rightAD/S=29.785kMedium/S\footnotesize\begin{split}v_2 &= 1,496\kern{-0,25em}\times\kern{-0,25em}10^8\, \rm km \cdot 1,991\kern{-0,25em}\times \kern{ -0,25em}10^{-7}\, rad/s\\&= 29,785\ \rm km/s\end{εˆ†ε‰²}v2​​=1.496Γ—108kilometerβ‹…1.991Γ—10-7rad/s=29.785km/s​

We're moving fast, aren't we?

Physical quantity that depends on angular velocity

Angular velocity is related to several physical quantities, some of which are listed below:

  1. angular acceleration– Describe how angular velocity varies with time. The larger the angular velocity difference, the larger the value of the angular acceleration. Feel free to see how it works in our practiceAngular Acceleration Calculator。

  2. rotational kinetic energy– Measurement of circular motion energy. As with kinetic energy, the (angular) velocity dependence is quadratic.

  3. centrifugal force- Feel how the vehicle feels while driving. The faster the turn, the sharper it is, the greater the centrifugal force, which we can clearly feel.

  4. Coriolis effect– This causes the object to rotate when on a rotating body (e.g. the Earth) instead of moving in a straight line.

  5. pulley system– This isn't actually a physical quantity, but it's an interesting device for dealing with angular velocity. The simplest system consists of two pulleys, usually with different circumferences or radii. A strap connects them, niceThey have the same linear speed, but due to their different sizes,Their angular velocities vary proportionally. Knowing this and having a motor with a well-defined rotational speed, we can set the angular velocity of the output element with high precision by simply changing the dimensions.

conservation of angular momentum

Some basic rules tell us how many quantities are conserved in isolated systems. The most famous are the conservation of energy and the conservation of momentum. with them alsoconservation of angular momentum. If we imagine two moments, we can write the rule as follows:

and1β‹…oh1=and2β‹…oh2,I_1 \cdot \omega_1 = I_2 \cdot \omega_2,and1​⋅oh1​=and2​⋅oh2​,

Whereand1i_1and1​Uand2i_2and2​are the initial and final moments of inertia of the mass; these dimensions describe the mass distribution around the center of the body.

We can see thatIncreasing moment of inertia decreases angular velocity and vice versa. So what are the consequences of this phenomenon?

Let's say you are a skater. When you spin, you have a certain angular velocity.When the hands are open, the moment of inertia is relatively large. Then bring your hands closer to the rest of your body. As a result, yourReduced moment of inertia, so since the total angular momentum must be conserved, yourangular velocity increase– which means you'll spin faster! It's not magic, it's physics!

If you can't/don't want to skate, you can try checking the rules with a regular swivel chair. Remember: safety first! Make sure you have enough space to run this experiment. Then start to rotate and watch how the angular velocity changes as you move the arm back and forth. Also, use some dumbbells for added effect. The result is that you can combine exercise and fun in one thing!


Is the angular velocity equal to vΓ—r?

born. Calculate the size of the angular velocityohLine speedvi gredaright, we separate these quantities:

Ο‰ = v / r

In this case, the units of angular velocity are equal torad/s(radial per second).

What is the angular velocity formula for constant angular acceleration?

fromAngular Velocity Formulain this case:

Ο‰ = Ο‰0+ Ξ±t


  • oh- angular velocity;
  • oh0– initial angular velocity;
  • A- angular acceleration; and
  • time– Event duration.

The disc is initially spinning at 27.5 rad/s with an angular acceleration of -10.00 rad/s2After 2 seconds the angular velocity is 7.5 rad/s:

Ο‰ = 27.5 rad/s + (-10.00 rad/s2) Γ— 2 seconds = 7,5 rad/s

How to convert RPM to rad/s

the bridgeConvert from RPM to rad/s:

  1. Use conversion factors:

    1 tpm = 0.10472 rad/s

  2. For example, if we wanted to convert 3500 RPM to rad/s, we would multiply it by the conversion factor:

    3500 tpm Γ— (0,10472 rad/s / 1 tpm)

  3. After multiplying we get:

    3500 tpm = 366,52 rad/s

Is the angular velocity equal to 2Ο€?

born, this expression is incomplete. We can calculate the magnitude of the angular velocity or angular frequency as the product of the factors2 pages(in radians, rad) and event frequencyF(in Hertz, Hz):

Ο‰ = 2pF


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