# What is Rotations: Definition and 193 Discussions

A rotation is a circular movement of an object around a center (or point) of rotation. The geometric plane along which the rotation occurs is called the rotation plane, and the imaginary line extending from the center and perpendicular to the rotation plane is called the rotation axis ( AK-seez). A three-dimensional object can always be rotated about an infinite number of rotation axes.
If the rotation axis passes internally through the body's own center of mass, then the body is said to be autorotating or spinning, and the surface intersection of the axis can be called a pole. A rotation around a completely external axis, e.g. the planet Earth around the Sun, is called revolving or orbiting, typically when it is produced by gravity, and the ends of the rotation axis can be called the orbital poles.

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1. ### I Physical reality of nontrivial loops in SO(3)

I will ask a mathematical and a physical-cum-philosophical question pertaining to the fact that SO(3) is not simply connected. Context Classical rotations in three spatial dimensions are represented by the group SO(3), whose elements represent 3D rotations. Having said that, note that classical...
2. ### I Not all reflections in 2D are 3D rotations?

Some reflections in the plane can be represented by a rotation in three dimensions, and some cannot: e.g., reflections across the x or y axes can. but a 2D reflection across the line x=y cannot. Thus the question in the summary.
3. ### MCNP6.2 - Combination of transformations

Hi everyone. I am struggling understanding how to combine more than one transformations, especially rotations. This stems mainly form the fact that it's unclear to me what reference frame is used to define the transformations angle if two consecutive transformations are applied. If I have a...
4. ### I Isomorphisms between C4 & Z4 Groups

0 Hint: Show that the isomorphism preserves the order of the element My solution: C4 = {e,r,r^2,r^3} where e-identity element and r is rotation by 90° Z4 = {0,1,2,3} LEMMA: ! Isomorphism preserves the order of the element ! (PROOF OF IT)Now we calcuate the order of the elements of both...
5. ### I The Mystery of Magnet Rotations: An Unanswered Question

This happened a long time ago, and I haven't found the answer. I posted a post on the Internet, and no one gave an explanation, so I really hope to find the answer. A magnet, no matter which pole is facing up, will rotate counterclockwise, as shown in the figure below. The magnet will not spin...
6. ### I Angular momentum and rotations

Cohen tannoudji. Vol 1.pg 702"Now, let us consider an infinitesimal rotation ##\mathscr{R}_{\mathbf{e}_z}(\mathrm{~d} \alpha)## about the ##O z## axis. Since the group law is conserved for infinitesimal rotations, the operator ##R_{\mathbf{e}_z}(\mathrm{~d} \alpha)## is necessarily of the form...
7. ### A Order of rotations due to torque in 3DOF in simulations

Hi, I am running a computational fluid dynamics (CFD) simulation. Supposed I have a symmetrical rigid body in space experiencing torque in the global x,y,z axes. It is stationary at t = 0. I also constrain it to only allow rotations in 3DOFs, and no translation. It will rotate and I need to...
8. ### I Experimenting with Spinor Rotations & Sign Changes

When a spinor is rotated through 360◦, it is returned to its original direction, but it also picks up an overall sign change. This sign has no consequence when spinors are examined one at a time, but it can be relevant when one spinor is compared with another. Is there an experiment to make an...
9. ### B What does it mean that a spin 1/2 particle needs two full rotations?

I know that we can change the spin orientation of a spin 1/2 particle up or down and test it in the Stern Gerlach apparatus. And the spin 1/2 particles need two full rotations to return to the previous state. Questions: 1). what does state mean? 2). Is, Changing spin orientation to up or...
10. ### I What Determines the Order of Rotations for a Gyro?

This could be a whole lot of nothing... however... Here are two figures used in gyroscopic analyses. On the left, is a model for an inertial guidance system on an airplane. As the airplane precesses (about the vertical 3-axis), and as the disk spins about the local 2-axis, there is an...
11. ### I Order of rotations: precession, nutation, spin

Hello I attach a picture of a problem from a dynamics textbook. The axle rotates about the axis AB WHILE (and the "while" here is a significant word to my question) it does that, the disk spins about an axis through C, but perpendicular to the face of the disk. As the textbooks solve...
12. ### A Exploring Infinitesimal Rotations in Classical Mechanics

Can anybody please help me to understand that why under infinitesimal rotation ##x1## transforms in the way as shown in equation 4-100? This is from Goldstein's Classical Mechanics page chapter 5 and page 168 on the Kinematics of Rigid body motion.
13. ### I to start with this Rotations problem

I have been stuck with this problem since the start of the week, and i don't know how i should start, any help is apreciated
14. ### The order of Euler Angle rotations for a top

Good Morning All. I have asked this before, but my post was not clear (my fault: I apologize). I hope this is more clear (please be patient as I try to get to the core of my confusion). In the first figure, below, the spinning top precesses as shown (well, it is not a animated jpg, but it...
15. ### Friction in Rotations: Do We Disregard It?

As far as I’ve gathered, for a system to rotate there has to be some static friction acting upon it and dynamic friction can be zero. But now I’m a bit confused about this as we completely disregarded static friction in some tasks where a system was rotating. So was my original assumption wrong...

41. ### I Finite and infinitesimal Rotations

Hi, I'm not sure about where I should post this question, so sorry in advance if I posted it in the wrong place. My question is basically this screenshot. So I really have some difficulty in understanding the two equations. I mean how can it not be equal? I understand that rotations are...
42. ### I Conceptual question about spin state rotations

My question is conceptual but specific. I'm self-studying Townsend's text 'A Modern Approach to Quantum Mechanics.' In Sec. 2.2 pg 33 (in case you have the book handy), he introduces rotation operators, in the context of spin states for spin-1/2 particles. He states that the rotation operator...
43. ### PID control for drone rotations

Hello, I'm playing around with simulating drones (quadcopters) in Gazebo (an open source robotics simulator). The control system is made up of six PIDs (one for each degree of freedom) and I'm encountering trouble tuning the pids for pitch / roll control. In this case, the linear x / y and...
44. ### A problem I couldn't solve -- Number of Earth rotations in a year....

< Mentor Note -- thread moved to HH from the technical physics forums, so the Homework Help Template is filled out farther down the thread > The number of rotations of Earth around its own axis in one year as measured by an observer from the sun.
45. ### Rotations from angular acceleration and final angular velocity

Homework Statement At a fair, Hank and Finn play with a horizontal 5.4 m long bar able to rotate about a pole going through its exact center. Hank pushes with 32 N at one end of the bar and Finn pushes with 18 N in the opposite direction at the other end. (Assume both forces are always...
46. ### Angular Velocity of a trebuchet

Hello, I need some help regarding angular physics. I am working on a project and I want to be able to predict (to some degree) the velocity of the payload leaving the trebuchet. (Excuse my ignorance I am just a high school student) Lets say a trebuchet see diagram has a counter weight m1...
47. ### Rotations around the x and y axes of stereographic sphere

Homework Statement Show that the equations $$\delta \phi = \cot \theta \cot \phi \delta \theta, \quad \delta \phi =- \cot \theta \tan \phi \delta \theta$$ represent rotations around the x and y axes respectively of a stereographic sphere. Both these two rotations map the sphere on itself and...
48. ### I Spherical coordinates via a rotation matrix

First, I'd like to say I apologize if my formatting is off! I am trying to figure out how to do all of this on here, so please bear with me! So I was watching this video on spherical coordinates via a rotation matrix: and in the end, he gets: x = \rho * sin(\theta) * sin(\phi) y = \rho*...
49. ### A Understanding the Connection Between Lie Algebras and Rotations

The Lie Algebra is equipped with a bracket notation, and this bracket produces skew symmetric matrices. I know that there exists Lie Groups, one of which is SO(3). And I know that by exponentiating a skew symmetric matrix, I obtain a rotation matrix. ----------------- First, can someone edit...
50. ### I Understanding Spinor Rotations

Hi, I am confused on a very basic fact. I can write \xi = (\xi_{1}, \xi_{2}) and a spin rotation matrix as U = \left( \begin{array}{ccc} e^{-\frac{i}{2}\phi} & 0 \\ 0 & e^{\frac{i}{2}\phi} \end{array} \right) A spinor rotates under a 2\pi rotation as \xi ' = \left( \begin{array}{ccc}...