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. F

    Vector to Tensor properties

    Homework Statement Suppose A and B are vectors. Show that the object Q with nine components Qij=AiBj is a tensor of rank 2. Homework Equations A tensor transforms under rotations (R) as a vector: Tij'=RinRjmTnm The Attempt at a Solution I wanted to just create the matrix, but I don't know how...
  2. polyChron

    Rotations in Bloch Sphere about an arbitrary axis

    Hey, (I have already asked the question at http://physics.stackexchange.com/questions/244586/bloch-sphere-interpretation-of-rotations, I am not sure this forum's etiquette allows that!) I am trying to understand the following statement. "Suppose a single qubit has a state represented by the...
  3. Strilanc

    I Mapping between rotations and operations: sign & handedness

    I have a toy quantum circuit simulator that I work on. I want to visually represent operations in multiple ways: as a Hamiltonian, as a unitary matrix, and as a Bloch sphere rotation. I want to double-check that I haven't flipped anything. I'll focus a concrete example: is this animation...
  4. S

    Insight/Intuition into rotations in R²

    I've been using rotation matrices for quite some time now without fully grasping them. Whenever I tried to develop an intuitive understanding of... x' = x\cos\theta - y\sin\theta \\ y' = x\sin\theta + y \cos\theta ... I failed and gave up. I've looked at numerous online texts and videos, but...
  5. M

    Finding number of rotations till arm breaks (circular motion)

    Homework Statement A 400 g steel block rotates on a steel table while attached to a 1.20 m -long hollow tube. Compressed air fed through the tube and ejected from a nozzle on the back of the block exerts a thrust force of 4.91 Nperpendicular to the tube. The maximum tension the tube can...
  6. L

    Fiction hypothetical about axial rotations

    Hello everyone! I am a fiction writer with a hypothetical question. For my work I have a new planet, the same size as Earth, with a moon with just a very slightly longer mean orbit (a few days). I have been reading about tidal locking which ensures that observers on the planet only see the...
  7. TheMathNoob

    Relation between complex eigenvalues and rotations

    Homework Statement I have the following matrix: 0 0 0 1 1 0 0 0 = A 0 1 0 0 0 0 1 0 and the vector v = (1,0,0,0) If I perform Av, this gives: Av=(0,1,0,0) And If I keep multiplying the result by A like A*A*(Av), the outcome will be something like j= (0,0,1,0) k=(0,0,0,1) l=(1,0,0,0) The...
  8. M

    Rotations in differential geometry

    Simple and basic question(maybe not). How are rotations performed in differential geometry ? What does the rotation matrix look like in differential geometry? Let us assume we have orthogonal set of basis vectors initially. I am looking to calculate the angle between two geodesics. Can this...
  9. L

    Number of wheel rotations relativistic car

    A car has wheels with exactly 1m of circumference and travels at constant relativistic speed with ##\gamma = 2## in an horizontal rectilinear road between two points A and B marked on the road itself and separated from a distance of 8 m, as measured from a frame of refence which is still with...
  10. W

    Rotations of spins and of wavefunctions

    This is a question regarding the intrinsic angular momentum S of a particle of spin 1. Assuming S = s(s+1)I = 2I and I is the identity operator. In our case s = 1. Let |z> be a ket of norm 1 such that Sz |z> = 0, and let |x> and |y> be the ket vectors obtained from it by rotations of + 1/2 Pi...
  11. H

    Rotations of 3-Spheres: Exploring 4D Space & Beyond

    I like to study rotations in higher dimensional spaces. It was worked out by Clifford in the late 19th century. A 3-sphere is a sphere in 4D space. A 3-sphere -- or any other object in 4space with non-zero volume -- can have two planes of rotation. These planes are always at right angles to...
  12. M

    Prove commutation relation of galilei boosts and rotations

    Homework Statement Use the formulas given (which have been solved in previous questions) prove that where w_12 is a complex constant. From here, induce that where eps_abc is the fully anti-symmetric symbol Homework Equations The equations given to use are: The Attempt at a...
  13. M

    Show that rotations and boosts lead to a combined boost

    Homework Statement Prove that applying an infinitesimal rotation of angle k<<1 around the axis x1, then a boost of speed -k along the axis x2 then the inverses of these is equal to a single boost of speed k^2 along the axis x3 The Attempt at a Solution Putting this into mathematical terms I...
  14. B

    Euler, Tait, Gyroscope: Rotations that cover it all

    I understand that a gyroscope undergoes precession, nutation, spin. And that the order of the rotations are such that the precession and spin share a common "local axis." I also understand there are, for totally different purposes, Euler angles to model rotations. In this case, the order of the...
  15. ShayanJ

    Invariance of the determinant under spin rotations

    Homework Statement Show that the determinant of a ##2 \times 2 ## matrix ## \vec\sigma \cdot \vec a ## is invariant under ## \vec \sigma\cdot \vec a \rightarrow \vec \sigma\cdot \vec a' \equiv \exp(\frac{i\vec \sigma \cdot \hat n \phi}{2})\vec \sigma\cdot \vec a \exp(\frac{-i\vec \sigma \cdot...
  16. R

    Camera on Airplane: Extrinsic plus Intrinsic 3D Rotations?

    Apologies up front for the long question … I have tried to be brief. I want to define camera angles for Google Earth (GE) when rotated about an aircraft yaw axis. The input is Latitude, Longitude, Altitude plus Heading, Pitch and Bank angles, actually coming from Flight Simulator. These drive...
  17. Adrian555

    Types of Rotations - Hi Everybody, Ask Your Questions Here

    Hi everybody, This is my first post, so I apologise for all the possible mistakes that I can make now and in the future. I promise that I'll learn from them! My question is the following: It's well-known the relationship between two pair of cartesian axes when a circular rotation is made...
  18. M

    Rotations in Bloch Sphere, and Free Parameters of a Qubit

    This question is mostly about group theory but I would like to understand it in the context of qubits rotating in a Bloch Sphere. What my understanding of things are right now: In the rotation Lie Group ##SO(3)##, we have three free parameters (##\frac{n(n-1)}{2}##), and this is also why we end...
  19. Gh778

    The position of a point in 2 rotations with 2 axes

    Homework Statement It's a question I ask to myself. A support turns at ##w_0## and can accelerate. A disk on the support can turn around itself (side view) but at start ##w_1=0##. I done the experimentation with 2 wheels but I'm not sure about my tests. There is an angle between 2 axes: The...
  20. uselesslemma

    Collisional excitation: selection rules for rotations?

    To be specific, I am referring to CO molecules undergoing collisions with H2, resulting in CO transitioning to an excited vibrational state. I can't seem to find any rotational selection rules for collisions, meaning ΔJ could be essentially anything, as long as energy and angular/linear momentum...
  21. stevendaryl

    Question about Coordinate Change

    Suppose that I have a two-dimensional coordinate system (x,y) and I change to a new coordinate system (u,v). What I know is that there is some function \theta(u,v) such that: \dfrac{\partial x}{\partial u} = cos(\theta) \dfrac{\partial x}{\partial v} = -sin(\theta) \dfrac{\partial y}{\partial...
  22. T

    Not sure about my rotations problem

    Homework Statement A rod of length L and mass M is balanced in a vertical position at rest. The rod tips over and rotates to the ground with its bottom attachment to the ground never slipping. Find the velocity of the center of mass of the stick just before it hits the ground and also find the...
  23. K

    Cable rotations formulae or method

    Hello, and thank you for your time. I observed a wire vibrating is good at blocking things trying to pass through their area. for example a rope spinning around can stop a tennis ball thrown through that area. obviously the speed of the rope allows it to "protect" an larger area than its gauge...
  24. F

    A question on the commutativity of finite rotations

    I was reading a section in my book discussing the commutativity of infinitesimal and finite rotations. In the book the authors try to set up a scenario to explain why finite rotations are not commutative. The following is an excerpt from the book regarding this language: "The impossibility of...
  25. S

    What is the general form of the rotation matrix in SU(2) space?

    Hi. I know that the \sigma matrices are the generators of the rotations in su(2) space. They satisfy [\sigma_i,\sigma_j]=2i\epsilon_{ijk}\sigma_k It is conventional therefore to take J_i=\frac{1}{2}\sigma_i such that [J_i,J_j]=i\epsilon_{ijk}\sigma_k . Isn't there a problem by taking these...
  26. Math Amateur

    MHB Rotations, Complex Matrices and Real Matrices - Proof of Tapp, Proposition 2.2

    I am reading Kristopher Tapp's book: Matrix Groups for Undergraduates. I am currently focussed on and studying Section 1 in Chapter2, namely: "1. Complex Matrices as Real Matrices".I need help in fully understanding the proof of Tapp's Proposition 2.2. Proposition 2.2 and its proof read as...
  27. Math Amateur

    MHB Rotations, Complex Matrices and Real Matrices

    I am reading Kristopher Tapp's book: Matrix Groups for Undergraduates. I am currently focussed on and studying Section 1 in Chapter2, namely: "1. Complex Matrices as Real Matrices".I need help in fully understanding what Tapp is saying in this section regarding the function \rho_n \ : \ M_n...
  28. V

    Crossed Belt and Rotational Dynamics: Solving for Angular Velocity and Torque

    Homework Statement http://imageshack.com/a/img908/282/eCETrh.png The green support turns clockwise at +w around the white fixed axis. The stator of the motor is fixed on the support. The stator of the generator is fixed on the support too. The motor drives the generator with a crossed belt...
  29. Erland

    Two errors about rotations in Goldstein's "Classical Mechanics"

    Classical Mechanics by Herbert Goldstein is one of the most used textbooks on this subject, perhaps the most used one. However, I found a couple of errors in Section 4.9 (in 3rd ed, written with Charles Poole and John Safko) about rotations. First, at p. 172, the angular velocity vector ω is...
  30. ChrisVer

    How is the Rotational Velocity of Galaxies Measured?

    I was wondering, how can we measure the rotational velocity of a galaxy? In practice knowing the mass distributions and so on, we could calculate it by classical mechanics (or maybe GR). However people measured the rotational velocity of the galaxies and found that it doesn't correspond to the...
  31. Strilanc

    Translating cumulative rotations into Pauli operators

    I want to write a program that, given the tracked position of a cube being rotated, applies analogous operations to a single qubit. The issue I'm running into is that, although operations correspond to rotations on the Bloch sphere, the mapping isn't one-to-one. So when I try to map back to...
  32. DiracPool

    Rotations in the complex plane

    I'm trying to check my understanding of rotations in the complex plane. Do I have any of this wrong? If so, can you please explain why? 1) Rotations a) Say we start with a vector, Q, defined on the real number line as (5,0). If I multiply that vector by i, we now have a vector "iQ" that...
  33. JonnyMaddox

    How can I calculate a rotation using geometric algebra?

    Hi, I want to calculate a rotation of a vector GA style with this formula e^{-B \frac{\pi}{2}}(2e_{1}+3e_{2}+e_{3})e^{B\frac{\pi}{2}}. Now since no book/pdf on GA exists where a calculation is explicitly done with numbers, I wounder how to calculate this. Should I substitude e^{-B...
  34. C

    How Do You Calculate Rotations and Translations in Geometry?

    Homework Statement 1. A group of American physicists works on a project where planar lines are in the form X=t⋅P+s⋅Q where P , Q are two fixed different points and s,t are varying reals satisfying s+t=1 . They need to know formulae for the images of the line X=t⋅P+s⋅Q in the following...
  35. skulliam4

    Rotations Per Minute Needed to Balance a Top

    I have little to no experience with this area of physics, so don't assume I know certain things. A cone is spun (with the tip down) at a constant, not decreasing, RPM (Rotations per Minute). What is the minimum RPM for it to stay there without falling, and (if possible) the minimum RPM necessary...
  36. G

    Active and Passive rotations

    Hi This is the problem... I'm reading a paper where the author says Transformation between the laboratory and spherical frames can be represented by the product of two rotations through the angles θ and \varphi, Û_{S}=\hat{R}_{y}(θ)\hat{R}_{z}(\varphi): |E>_{S}=Û_{S}|E>_{L}. The matrix...
  37. M

    Why do infinitesimal rotations commute but finite rotations do not?

    In K&K's Intro to Mechanics, they kick off the topic of rotation by trying to turn rotations into vector quantities in analogy with position vectors. It's quickly shown, however, that rotations do not commute, making them rather poor vectors. They then show, however, that infinitesimal rotations...
  38. E

    Composition of collateral rotations of a planet

    A body is orbiting the sun and rotates about its axis (z). My coordinate system is co-rotating with the body. I need to determine how does a vector that points to the sun change after a certain period of time. Initially the sun vector lies in the xz plane. Basically I need to find the rotation...
  39. R

    Applying 3 Concurrent Angular Velocities to Vectors

    This problem has me stumped. I'm toying with a stabilization platform design which has 3 gyroscopes supplying angular velocity -- one for each axis (x,y,z). The model has units vectors (x,y,z) representing the platform's orientation in space. The question is how do I apply the 3 orthogonal...
  40. S

    3D Rotations using complex numbers

    I was thinking that if you could use quaternions to rotate an object using quaternion algebra that there might be a way to rotate an object using complex numbers in some fashion. I was looking at quaternion rotation of a vector and the amount of operations seemed to be a lot. Of course it levels...
  41. K

    MHB Determine the matrices that represent the following rotations of R^3

    I need to determine the matrix that represents the following rotation of $R^3$. (a) angle $\theta$, the axis $e_2$ (b) angle $2\pi/3$, axis contains the vector $(1,1,1)^t$ (c) angle $\pi/2$, axis contains the vector $(1,1,0)^t$ Now, I would like to check if I got the right answers because...
  42. R

    Quaternions & Rotations in 3D Space: What You Need to Know

    Hey there, I have a question regarding quaternions and rotations. It's related to 3D graphics programming, but nevertheless I'm sure the physics forum is the right place for my question. As far as I've understood there are 3 primary ways we can store rotation: euler angles, quaternions and...
  43. M

    2 Rotations on different coordinate systems

    I have a question that i been trying to solve which seam simple but been having trouble. Today I thought about rotation matrix and how the following problem would be solved. Initial Coordinate system (x,y,z) a rotation is desired about x let's say α=30 degrees so that a new coordinate...
  44. A

    Doubt regarding rotations in spinor space

    Dear Sir, I am currently doing an advanced course in Quantum Mechanics. This current doubt of mine, I am unable to clarify it properly. It follows as: Spin 1/2 particles reside in 2dim-Hilbert space( Spinor Space)...However, we talk about rotations of states in this space where the angle...
  45. N

    Euclidian and Hyperbolic rotations

    Do hyperbolic rotations of euclidian space and ordinary rotations of euclidian space form a group?
  46. N

    Obtaining spherical coordinates by rotations

    Hi Say I have a point on a unit sphere, given by the spherical coordinate $(r=1, \theta, \phi)$. Is this point equivalent to the point that one can obtain by $(x,y,z)=(1,0,0)$ around the $y$-axis by an angle $\pi/2-\theta$ and around the $z$-axis by the angle $\phi$? I'm not sure this is...
  47. D

    Mapping Rotations of Relative Axes to Fixed Axes

    So I have been trying to figure out some orientation data that I gather from a triaxial gyroscope, and figure out my orientation using only initial conditions and angular velocity from the gyroscope's current axes. The data is all relative to the current orientation, so if I rotate the device...
  48. J

    3D Rotations - Confused with multiple rotations

    I've been trying to wrap my head around 3D rotations and specifically, Euler angles. I thought I had a good understanding of performing multiple rotations, but after writing a Matlab script that graphically shows my rotations, I am not so sure right now. My biggest hang up is whether or not a...
  49. K

    How to Convert Rotations per/sec to radians per second

    Homework Statement How do you convert Rotations per/sec to radians/sec Ex: a disk rotates 15 times in 25 seconds convert to radians/sec Attempt: 15 rotations / 25 seconds = 1 rotation / 0.6 seconds 1 rotations = 2 pi rad
  50. Fredrik

    Geometric insight about rotations needed

    I'm trying to find a way to simplify a complicated proof. The worst step of the proof involves a product of five 4×4 matrices. I'm hoping, perhaps naively, that if I could understand why the result of this operation is so simple, I may be able to explain the proof to others without actually...
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