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- Thread starter Sundown444
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Nugatory

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Not really. Please explain it to me.

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CWatters

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I believe rotation about two axis at once is equivalent to rotating about a third axis alone. So anything rotating about a single axis (like a car wheel) can be said to be rotating about two other axis at once.

The only analogy I can come up with is something sliding down a slope. You can break down its motion into horizontal and vertical components.

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CWatters

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A.T.

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What does "rotate in two different axes" mean mathematically? You can decompose an angular velocity vector into arbitrary components.First, is it possible for someone or something to use torque to rotate in two different axes of rotation at once?

Yes, see for example:In a separate situation, is it possible to tilt using torque to tilt the axis of rotation to rotate in a different angle?

https://en.wikipedia.org/wiki/Gyroscope

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What does "rotate in two different axes" mean mathematically? You can decompose an angular velocity vector into arbitrary components.

Yes, see for example:

https://en.wikipedia.org/wiki/Gyroscope

Mathematically? I am not sure how to explain it well. Best I can say is that, for example, a gymnast does a twist and flip at the same time.

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russ_watters

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Let me repeat what @A.T. said in reverse: Any two arbitrary components of angular velocity can be combined into one angular velocity vector.Mathematically? I am not sure how to explain it well. Best I can say is that, for example, a gymnast does a twist and flip at the same time.

In other words, any rotating object can be said to be rotating about one axis or several, depending on how you want to slice it.

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Let me repeat what @A.T. said in reverse: Any two arbitrary components of angular velocity can be combined into one angular velocity vector.

In other words, any rotating object can be said to be rotating about one axis or several, depending on how you want to slice it.

One axis or several? Forgive me, but is it really both that can happen? I'd appreciate it if you explain it more because I am confused. I think I get everything else about the two angular velocity components part, though.

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russ_watters

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Do you know how a diagonal line can be broken into horizontal and vertical components? It's like that.One axis or several? Forgive me, but is it really both that can happen?

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Do you know how a diagonal line can be broken into horizontal and vertical components? It's like that.

So it is more of a gymnast rotating in a diagonal direction?

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russ_watters

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https://ocw.mit.edu/courses/aeronau...fall-2009/lecture-notes/MIT16_07F09_Lec25.pdf

You can also google "adding angular velocity vectors". There are lots of detailed explanations, including youtube videos.

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https://ocw.mit.edu/courses/aeronau...fall-2009/lecture-notes/MIT16_07F09_Lec25.pdf

You can also google "adding angular velocity vectors". There are lots of detailed explanations, including youtube videos.

I might get this wrong again, but is it anything like this?

http://www.physicsclassroom.com/class/vectors/Lesson-1/Vector-Addition

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russ_watters

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Sure, but recognize that you can rotate your coordinates and call any direction "diagonal". The main point here was to answer your original question. The answer is that you only have one actual axis at a time, but can break it apart or combine it mathematically for the purpose of analysis. E.G., if you apply a torque in one direction and then a torque in another direction, it will allow you to find the final axis of rotation.So it is more of a gymnast rotating in a diagonal direction?

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russ_watters

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Yes, that was my example. I used it because it is easier to visualize. My point was that rotational motion vectors also can be added together.I might get this wrong again, but is it anything like this?

http://www.physicsclassroom.com/class/vectors/Lesson-1/Vector-Addition

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Yes, that was my example. I used it because it is easier to visualize. My point was that rotational motion vectors also can be added together.

So when the gymnast, diver or whatever appears to be rotating in two axes at once, they are actually moving in one, combined axis of rotation rather than two different ones?

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A.T.

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A gymnast is not a rigid body, so there might not be a unique angular velocity vector. But there still is a unique total angular momentum vector. See for example the falling cat:So it is more of a gymnast rotating in a diagonal direction?

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russ_watters

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Yes.So when the gymnast, diver or whatever appears to be rotating in two axes at once, they are actually moving in one, combined axis of rotation rather than two different ones?

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jbriggs444

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If you take one rotation and a second rotation, you can combine them. The rotations are representable as matrices. You multiply a vector by the rotation matrix to obtain a new vector. You can combine two rotation angles by performing a matrix multiplication on the rotation matrices. In this sense, rotations "multiply" rather than "add".Yes, that was my example. I used it because it is easier to visualize. My point was that rotational motion vectors also can be added together.

$$\vec{v_f} = ( \vec{v_i} \times R_1 ) \times R_2 = \vec{v_i} \times (R_1 \times R_2)$$

If you are dealing with infinitesimal rotation angles you will be dealing with a rotation matrix that is only infinitesimally different from the identity matrix. If you multiply two of these together you have something that is very similar to ##(1+\theta)\times (1+\gamma) = 1 + \theta + \gamma + negligible)##. This is how one can speak of rotations adding when they are actually multiplying.

Caveat: Linear algebra and three dimensional rotations are not my forte. No formal training here.

Edit: Note that a rotating rigid body will not always rotate in a regular fashion around an unchanging axis. If you let it rotate through 360 degrees, it will not, in general, return to its original orientation. Its instantaneous motion can always be characterized as a rotation around a particular axis. But its continued motion may involve rotation around a precessing progression of different axes. [I first convinced myself of this many years ago, tossing a pencil in the air with a combination of an end over end rotation and a spin around the long axis].

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CWatters

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Yes. Although people can bend which means different parts of them can rotate about different axis which confuses things. Try playing with something rigid like a football.So when the gymnast, diver or whatever appears to be rotating in two axes at once, they are actually moving in one, combined axis of rotation rather than two different ones?

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Okay, thanks everyone!

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They appear to be rotating in two axes at the same time in some parts of the video.

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A.T.

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They appear to be rotating in two axes at the same time in some parts of the video.

What do you mean by "rotating in two axes"? That the spin axis is not aligned with any of anatomic body axes? It doesn't have to be, even for rigid bodies. And humans aren't event rigid, so how do you define the axes? (see post #18).

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What do you mean by "rotating in two axes"? That the spin axis is not aligned with any of anatomic body axes? It doesn't have to be, even for rigid bodies. And humans aren't event rigid, so how do you define the axes? (see post #18).

Could you explain the "spin axis is not aligned with any of anatomic body axes" more?

What I meant, is that the divers seem to be doing somersaults and twists at the same time. If I have this down right, I wonder how it is possible?

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jbriggs444

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It is easy to think of the human body rotating forward about the waist (a series of handsprings or forward rolls). It is easy to think of the human body rotating around its vertical axis (a ballerina or an ice skater doing a pirouette). It is easy to think of the human body rotating clockwise or counterclockside about the waist (doing cartwheels). A rigid rotation about any other axis will tend to look funny and will likely not be balanced.Could you explain the "spin axis is not aligned with any of anatomic body axes" more?

Any instantaneous rigid motion can be characterized in terms of rotation about one axis, not two. However, there are two complicating factors.What I meant, is that the divers seem to be doing somersaults and twists at the same time. If I have this down right, I wonder how it is possible?

1. Divers are not rigid.

2. Even for rigid motion, the axis of rotation need not remain fixed.

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russ_watters

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Why would you think it not possible? Surely not based on prior discussion in the thread...?What I meant, is that the divers seem to be doing somersaults and twists at the same time. If I have this down right, I wonder how it is possible?

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It is easy to think of the human body rotating forward about the waist (a series of handsprings or forward rolls). It is easy to think of the human body rotating around its vertical axis (a ballerina or an ice skater doing a pirouette). It is easy to think of the human body rotating clockwise or counterclockside about the waist (doing cartwheels). A rigid rotation about any other axis will tend to look funny and will likely not be balanced.

Any instantaneous rigid motion can be characterized in terms of rotation about one axis, not two. However, there are two complicating factors.

1. Divers are not rigid.

2. Even for rigid motion, the axis of rotation need not remain fixed.

So, since divers are not rigid, they can apparently rotate on more than one axis? Sorry if I have this down wrong.

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jbriggs444

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Because they are not rigid, the notion of rotating about an axis is nonsensical in the first place.So, since divers are not rigid, they can apparently rotate on more than one axis? Sorry if I have this down wrong.

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Because they are not rigid, the notion of rotating about an axis is nonsensical in the first place.

So, they can rotate around any axis/number of axes, then?

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