# Angular velocity in dimensional analysis

• mightysteve
In summary, angular velocity is a measure of how fast an object is rotating around an axis and is expressed in terms of radians per second or revolutions per second. Radians are a dimensionless measure and the 'rad' in 'rad/s' is used to specify angular velocity. Revolutions, on the other hand, are counted in pure basic numbers and therefore, are also dimensionless.

#### mightysteve

Hullo was wondering if anyone could help me. In dimensional analysis using then buckingham pi theorum, I'm not sure how to express an angular velocity in terms of basic dimensions (i.e M (mass), L (length), T(time), $$\Theta$$ (temp).

I know an angular velocity is revs/s or rad/s so its going to be 'whatever radians/ revs are in basic dimensions * T^(-1)

But I'm not sure how to express revs/rads in basic dimensions? Are they just dimensionless?

Yes. Radian is a dimensionless measure. If there is a circle, and a small part of it's circumference [an arc] subtends an angle $\theta$ radians at the center, then $\theta$ is the ratio of the length of the arc to the radius of that circle. As you can see, a radian is a ratio of two lengths: the arc length and the radius and hence, it is dimensionless as the length dimension of both these quantities cancel each other out.

The 'rad' in 'rad/s' is there to denote that we specifically mean a unit of angular velocity, to separate it from '/s', which is a unit of frequency. However, angular velocity is also the angular frequency of rotation in a uniform circular motion i.e. it differs from the no. of rotations by a factor of $2\pi$.

As far as revolution is considered, n revolutions just mean $n \times 2\pi$ radians. Even then, revolutions are counted in pure basic numbers. And as you know, there is nothing like a 'kilo numbers' or a 'mega numbers'.. hence revolutions is also a dimensionless unit.

Aah, i see. Thanks very much

## 1. What is angular velocity in dimensional analysis?

Angular velocity in dimensional analysis is a measurement of the angular displacement of an object per unit time. It is often denoted by the symbol ω and is commonly measured in radians per second.

## 2. How is angular velocity different from linear velocity?

Angular velocity is a measurement of how quickly an object is rotating around an axis, while linear velocity is a measurement of how quickly an object is moving in a straight line. Angular velocity takes into account the distance from the axis of rotation, while linear velocity does not.

## 3. What are the units of angular velocity?

The units of angular velocity are typically radians per second (rad/s), although degrees per second (°/s) can also be used. Dimensionally, it can be expressed as length over time, such as meters per second (m/s).

## 4. How is angular velocity calculated?

Angular velocity is calculated by dividing the change in angular displacement by the change in time. It can also be calculated by dividing the linear velocity by the radius of the circle it is traveling in.

## 5. Why is angular velocity important in physics?

Angular velocity is important in physics because it is used to describe rotational motion and is a key component in understanding concepts such as torque, centripetal force, and angular momentum. It is also crucial in fields such as engineering, astronomy, and mechanics.