# Sperical coords: Position vector of spinning disk.

• vwishndaetr
In summary, the position vector of a point on a rotating wheel in spherical coordinates is (R, \frac{\pi}{2}, \frac{\pi}{2}); the angular velocity is quadratic, so the disc is accelerating, so the position should be third order correct.
vwishndaetr
I posted this in the Intro physics sections, but then realized that spherical coords might be a bit complex for introductory physics. This has been bothering my head for a couple days now. Any help is appreciated.

Given: A wheel of radius R rotates with an angular velocity. The wheel lies in the xy plane, rotating about the z-axis.

$P(x,y,z) = (0,R,0)$

$$\overrightarrow{\omega}= Ct^2\hat{k}$$

Ques: What is the position vector of point P in spherical coordinates?

Ans: Now I know that,

$$P(r,\theta,\phi,) = (R,\frac{\pi}{2},\frac{\pi}{2})$$

But I don't think that helps much.

For the position vector, I can't figure out the term for:

$$\hat{\phi}$$

I have:

$$\overrightarrow{r}= R\ \hat{r}+\frac{\pi}{2}\ \hat{\theta}+\ \ \ \ \ \ \ \ \hat{\phi}$$

The last term is giving me issues.

Now I know that $$\phi$$ changes with time, so the term must depends on $$t$$.

I also know that $$\omega$$ is $$rad/s$$, which can also be interpreted as $$\phi/s$$.

But I don't think it is legal to just integrate $$\omega$$ to get position. Is it?

Since the angular velocity is quadratic, that means the disc is accelerating. So the position should be third order correct?

I'm being really stubborn here because I know it is something minute that is keeping me from progressing.

Why wouldn't it be legal to integrate? You have already established that $$|\mathbf{\omega} (t)| = \dot{\phi}(t)$$

So $$\phi (t) = Ct^3/3 + \pi / 2$$

Maybe I am missing something here but it looks straightforward..

I don't know, just seemed out of place. Since in sperical coords the formula from position to velocity pics up an R for the phi-term, it throws me off.

Edit: Just realized that what I just said relates to tangential velocity. So it would pick up an R.

Thank you for clarifying this for me. I lacked the confidence to make that jump.

Yes
$$\mathbf{v} = \mathbf{\hat{r}}\dot{r} + \mathbf{\hat{\theta}} r \dot{\theta} + \mathbf{\hat{\phi}}r\dot{\phi}\sin{\theta}$$

In your example
$$\dot{r} = 0, \quad r = R, \quad \dot{\theta} = 0, \quad \theta = \pi / 2 , \quad \dot{\phi} = |\mathbf{\omega}|$$
so
$$\mathbf{v} = R\mathbf{\hat{\phi}}\dot{\phi} = R\mathbf{\hat{\phi}}|\mathbf{\omega}|$$

Thanks again!

Appreciate it.

## 1. What is the position vector of a spinning disk in spherical coordinates?

The position vector of a spinning disk in spherical coordinates is a mathematical representation of the location of a point on the disk relative to a fixed reference point, using the distance from the reference point, the polar angle, and the azimuthal angle.

## 2. How are spherical coordinates used to describe the motion of a spinning disk?

Spherical coordinates are useful in describing the motion of a spinning disk because they allow for a more intuitive understanding of the orientation and position of the disk. By using the polar and azimuthal angles, we can easily visualize the rotation and movement of the disk.

## 3. What are the advantages of using spherical coordinates to describe a spinning disk?

One advantage of using spherical coordinates is that they are well-suited for describing circular or rotational motion. Additionally, they can simplify complex calculations by reducing the number of variables needed to describe the position and orientation of the spinning disk.

## 4. Can the position vector of a spinning disk be expressed in other coordinate systems?

Yes, the position vector of a spinning disk can also be expressed in other coordinate systems, such as Cartesian coordinates or cylindrical coordinates. However, spherical coordinates are often preferred for their simplicity and ability to describe rotational motion.

## 5. How are spherical coordinates related to other coordinate systems?

Spherical coordinates are related to other coordinate systems through mathematical transformations. For example, they can be converted to Cartesian coordinates using trigonometric functions. Additionally, they can be transformed into cylindrical coordinates by setting the polar angle to 90 degrees.

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