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Why is angular velocity the same for all points on a spinning disk, even though they are at different radii from the center?

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  • $\begingroup$ I've removed a number of comments that were attempting to answer the question and/or responses to them. Commenters, please keep in mind that comments should be used for suggesting improvements and requesting clarification on the question, not for answering. $\endgroup$
    – David Z
    Commented May 3, 2020 at 11:16
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    $\begingroup$ Question titles should be like book titles. They distinguish the book from other books, give a hint about the content, but do not tell the full story. The content then should elaborate the topic to full depth the author is able to do. $\endgroup$
    – Poutnik
    Commented May 3, 2020 at 12:07
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    $\begingroup$ Question titles should not be like book titles. They should give as much information about the question as possible within the space limits. This title does not seem too long to me. $\endgroup$
    – JiK
    Commented May 3, 2020 at 14:25
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    $\begingroup$ Because the concept of angular (versus linear) velocity was introduced in order to describe the phenomenon of a rotating rigid disk $\endgroup$ Commented May 3, 2020 at 18:14

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Because angular velocity is measured in radians per second. Every point on a spinning disk along a radial line from the centre completes one full revolution ($2\pi$ radians) in exactly the same amount of time. You are confusing tangential velocity with angular velocity, tangential velocity is different at every point along the same radial line.

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Omega

The angular velocity $\omega$ is quite literally the rate of change of the angle $\theta$:

$$\omega=\frac{\text{d}\theta}{\text{d}t}$$

Clearly it is independent of radius.

The tangential velocity of the point $P$ however is dependent on the radius:

$$v_P=\omega R(P)$$

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Yes. Angular velocity is a property of the body or the reference frame and it does not depend on the location where it is measured.

Since it can be defined without any location information, the rotational velocity vector $\boldsymbol{\omega}$ is pure vector conveying only magnitude and direction.

This is to contrast with translational velocity $\boldsymbol{v}$, which must be defined at a position to have meaning. And in general translational velocity varies by location.

$$\boldsymbol{v}_A = \boldsymbol{v}_B + \boldsymbol{\omega} \times \boldsymbol{r}_{A/B} $$

The exception being a pure translation where all the points on a body have the same value. But that is just a special case.

In fact, to know where a body is rotating about (the axis of zero velocity) you need to consider both rotational and translational velocity.

$$\boldsymbol{r}_{\rm axis} = \frac{ \boldsymbol{\omega} \times \boldsymbol{v} }{ \| \boldsymbol{\omega} \|^2 } $$


Momentum $\boldsymbol{p}$ is another pure vector, where a body has momentum and there is no need for defining the location of where momentum is measured.

Angular momentum $\boldsymbol{L}$, on the other hand, needs location specification, you measure it at different points using a similar law

$$\boldsymbol{L}_A = \boldsymbol{L}_B + \boldsymbol{p} \times \boldsymbol{r}_{A/B} $$

Similarly the line of action of momentum ,where angular momentum is zero is found when considering both linear and angular vectors at the same time.

$$\boldsymbol{r}_{\rm axis} = \frac{ \boldsymbol{p} \times \boldsymbol{L}}{\| \boldsymbol{p} \|^2} $$

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To understand the reason, one should be knowing that the angular velocity of a point on a rotating body is the angle swept per unit time by the straight line joining that point to the center of rotation.

It is worth noticing that the line joining each point on a spinning disk to the center of rotation/spin sweeps an equal angle in equal time interval irrespective of the radial distance of concerned point on spinning disk. Therefore the angular velocity remains constant for each and every point on a spinning disk.

Let there be a disk spinning about its center O. Consider any two arbitrary points say A & B at radial distances $r_1 $ & $r_2$ respectively such that at start of spinning, the angle between the line OA & X-axis is $\alpha$ & angle between lines OA & OB is $\beta$. Now let the disk spin uniformly through an angle $\theta$ over time $t$ .

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Then the angular velocities of points A & B are given as

$$\omega_A=\frac{\text{Angle swept by line OA}}{\text{Time taken}}=\frac{\alpha+\theta-\alpha}{t}=\frac{\theta}{t}$$
$$\omega_B=\frac{\text{Angle swept by line OB}}{\text{Time taken}}=\frac{\alpha+\beta+\theta-(\alpha+\beta)}{t}=\frac{\theta}{t}$$ Thus the angular velocities of points A & B at different radial distances are equal i.e. $\omega_A=\omega_B$

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Everybody standing on a turntable will be rotating at the same rate (they will face the north wall the same number of times every second, for example) even though they are moving at different speeds.

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If you also think about it intuitively, you will notice that the angle traveled for an object along a circular object is will the same no matter how far or close it is to the center. It is true that the arc length changes as the radius increases, however, that is what is known as tangential velocity. So since the radius has no play in the size of an angle, angular velocity doesn't change because of the radius.

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