# B Torque and Rotation (1 Viewer)

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#### Sundown444

I have some questions about torque and its role in gymnastics, or with anything that may rotate, possibly. First, is it possible for someone or something to use torque to rotate in two different axes of rotation at once? In a separate situation, is it possible to tilt using torque to tilt the axis of rotation to rotate in a different angle? And if yes is the answer for either question, please explain to me: how is it possible?

#### Nugatory

Mentor
The answer to both questions is "yes". But before we go any further.... are you familiar with the definition of torque as the cross-product of two vectors: $\vec{T}=\vec{r}\times\vec{F}$?

#### Sundown444

The answer to both questions is "yes". But before we go any further.... are you familiar with the definition of torque as the cross-product of two vectors: $\vec{T}=\vec{r}\times\vec{F}$?
Not really. Please explain it to me.

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#### Sundown444

It's been around two days. Is anyone going to answer the question, let alone the "how is it possible part"? I hope it is okay for me to make this post even though the last post before is mine.

#### CWatters

Homework Helper
Gold Member
I'm a bit rusty but...

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.

#### CWatters

Homework Helper
Gold Member
Are you really asking about rotation about a moving axis? Sometimes people see something rotating about a moving axis and think it's rotating about two axis at the same time.

#### A.T.

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

In a separate situation, is it possible to tilt using torque to tilt the axis of rotation to rotate in a different angle?
Yes, see for example:
https://en.wikipedia.org/wiki/Gyroscope

#### Sundown444

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.

#### russ_watters

Mentor
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.
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.

#### Sundown444

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.

#### russ_watters

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

#### Sundown444

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?

• CWatters

#### russ_watters

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

Mentor

#### Sundown444

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?

• CWatters

#### A.T.

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

#### russ_watters

Mentor
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?
Yes.

#### jbriggs444

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.
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".

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