# What is the angular velocity in the center of a rotating disc?

• Seth Greenberg
In summary, the conversation discusses the motion of a disc and its angular velocity at different points on the disc. It is mentioned that the linear velocity is dependent on the position vector and is zero on the axis of rotation, but different points on the disc have the same angular velocity. The conversation also mentions the non-classical case, where the linear velocity is not zero but very small when the position vector is very small.
Seth Greenberg
I have a disc. The center of the disc is its center of mass and the motion of the disc is purely rotational (no translation). What is the angular velocity in the center of the rotating disc?

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Is this a statement or is there a question in there somewhere?

What is the angular velocity in the center of the rotating disc?

Assuming that the disk is a rigid body, its angular velocity is ##\vec \omega## at its center and at any other point on it. It is the linear velocity ##\vec v=\vec \omega \times \vec r## that depends on the position vector ##\vec r## and is zero on the axis of rotation. Different points on the disk have different linear velocities but the same angular velocity about the axis of rotation.

By 'linear velocity' you mean tangential velocity? What happens if ##\vec{r}## is not zero but very, very small, say the plank length ##\ell_P## and ##\vec{\omega} = 1##?

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Seth Greenberg said:
By 'linear velocity' you mean tangential velocity? What happens if ##\vec{r}## is not zero but very, very small, say the plank length?
If ##\vec \omega## is constant, as I assume to be the case here, "tangential" and "linear" velocity are the same. When ##\vec{r}## is not zero but very, very small, then the linear velocity classically is not zero but very, very small.

What is the non-classical case?

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## 1. What is angular velocity?

Angular velocity is a measure of the rate at which an object rotates around a fixed axis. It is typically measured in radians per second or revolutions per minute.

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

Angular velocity is a measure of rotational speed, while linear velocity is a measure of straight-line speed. Angular velocity takes into account the distance from the axis of rotation, while linear velocity does not.

## 3. How is angular velocity calculated?

Angular velocity can be calculated by dividing the change in angle by the change in time. It can also be calculated by dividing the linear velocity by the radius of the rotating object.

## 4. What is the unit of measurement for angular velocity?

The unit of measurement for angular velocity is radians per second (rad/s) or revolutions per minute (rpm).

## 5. How does angular velocity affect the behavior of a rotating disc?

Angular velocity determines the speed at which the disc rotates, as well as the direction of rotation. It also affects the centripetal force and the stability of the disc as it rotates.

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