Rotations Definition and 193 Threads

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. M

    How many rotations of peddels per second?

    Homework Statement A bicycle drives on a straight road with a velocity v = 18 km/h. How many turns (rotations) of peddles per second must be carried out by cyclists, if the radius of the wheels r = 35 cm, and the ratio of the radii of front and rear gear wheels is 2:1? Homework Equations I...
  2. P

    Sum of infinitesimal rotations around different points in 2D space ?

    Hello, in order to numerically solve a physics problem I think I need to add 2 (infinitesimal) rotations of one and the same segment each around a different point in 2D space in one iteration of numeric approximization. How does this addition work out? Is it the sum of the vectors connecting...
  3. F

    Comparing Planetary Orbits & Rotations in Pictured Figures

    I want to know the difference between the 2 types of rotations shown in the picture attached . planetary orbiting are similar to the second figure in the picture attached , all i want to know what is the difference between the 2 figures .
  4. A

    Generator of rotations rotations of WHAT?

    "Generator of rotations"...rotations of WHAT? The angular momentum operator is defined as the "generator of rotations." Fine. But rotations of WHAT? What's being rotated? The wave function (doesn't make sense; isn't the wave function a scalar), perhaps? An arbitrary vector in the coordinate...
  5. L

    Calculating Earth's # of Rotations Around the Sun

    I'm doing physics of uniform circular motion, while I'm working out this equation It occurred to me , is it possible to find out how many rotations of Earth around the sun until it hits the sun. By the way I'm at a very NOOB level lol but seriously? *thinks of possible equation* :( not...
  6. G

    Defining Addition Operator for Algebra of Rotations in d-Dimensions

    Suppose I have objects which are rotations of d-dimensional real vectors with an additional optional scaling. Concatenating means multiplication of these objects. I want to define an addition operator, so that the "sum" of two rotations gives another unique rotation with scalings only. Which...
  7. A

    Composition of Lorentz pure rotations

    Hello, Given (in spherical coordinates) the resulting 4-vector K of the composition of 2 Lorentz pure rotations R1 and R2, where only R1 is known, I would like to find the angle of the "overall" rotation resulting from this composition. In other words, I want to find the symbolic expression...
  8. B

    Why do rotations and reflections behave differently in R^2 and R^3?

    In R^2, a reflection can be achieved equally well by a rotation, because the group space of U(1) is connected. Visually I think of this as being able to peform a rotation that moves the positive y-axis into the negative y-axis and vice versa and without changing the x-axis by imagining R^2 as...
  9. T

    Calculating Torque and Rotational Inertia for Opening a Heavy Door

    Homework Statement You need to open a heavy door that's 30 inches wide. Instictively, you push near the edge that's farthest from the hinges. If, instead, you had pushed at a point only 10 inches in from the hinges, how much harder would you have had to push to open the door at the same speed...
  10. N

    Rot Matrix for nxn: Group of Rotations

    In general, what is the rot matrix for nxn?
  11. F

    Circluar Motion Involving Rotations Per Minute

    Homework Statement What is the magnitude of the acceleration of a speck of clay on the edge of a potter's wheel turning at 40 rpm (revolutions per minute) if the wheel's diameter is 35 cm? Homework Equations v=2(pi)(r)/t a=(v^2)/r The Attempt at a Solution 2(3.14)(.175)/.67 =...
  12. L

    Moments of Inertia, Kinetic Energy and rotations

    Homework Statement A bicycle racer is going downhill at 11.0 m/s when, to his horror, one of his 2.25 kg wheels comes off when he is 75.0 m above the foot of the hill. We can model the wheel as a thin-walled cylinder 85.0 cm in diameter and neglect the small mass of the spokes. How fast is...
  13. S

    Proving there is a fixed point in a discrete group of rotations

    Homework Statement Let G be a discrete group in which every element is orientation-preserving. Prove that the point group G' is a cyclic group of rotations and that there is a point p in the plane such that the set of group elements which fix p is isomorphic to G' The Attempt at a...
  14. M

    Conservation of Angular Momentum in Collision with Rotations

    A uniform rod of length L1 = 2.2 m and mass M = 2.8 kg is supported by a hinge at one end and is free to rotate in the vertical plane. The rod is released from rest in the position shown. A particle of mass m is supported by a thin string of length L2 = 1.8 m from the hinge. The particle sticks...
  15. L

    Understanding Pauli Matrices and Rotations

    I have some questions about Pauli matrices: 1. How do we calculate them? Which assumptions are needed? Are the assumptions related to properties of orbital angular momentum in any way? 2. How do we prove that the Pauli matrices (the operators of spin angular momentum) are the generators...
  16. C

    How Do You Perform a 90 Degree Rotation of a Complex Number Around a Point?

    I'm reading this book on modern geometry and I was wondering if I'm doing these problems right: if I'm give a point 2+i and I'm suppose to rotate is 90 degrees first I move it to the origin T(z)=z-(2+i) second, I rotate it e^(pi/2*i)*z I'm not sure how to interpret that...
  17. P

    Understanding the D^{l}(\theta) Representation of 3D Rotations

    I'm having difficulty with the D^{l}(\theta) representation of 3D rotations what do the mean i suppose one you construct it for l = 1 you get the general rotation Euler matrix for 3-d Space, but what do the l = other integers or half integers mean physically? is the D matrices the...
  18. M

    Givens rotations versus Euler angles

    Trying to implement QR decomposition using Givens rotations, I calculate G1 to zero n32 of original matrix A, then G2 to zero n31 of G1 * A, then G2 to zero n21 of G2 * G1 * A. Residual matrix, R = G3 * G2 * G1 * A comes upper triangular as expected, so I believe my code is correct. Looking...
  19. P

    Rotations of Earth and other Rigid Bodies

    Does anyone know of a good mathematical reference covering the use of one-parameter group actions to model rotations of planets and/or other rigid bodies? Thanks in advance!
  20. Philosophaie

    Are All Planets Rotating in the Same Direction, Except Venus and Uranus?

    I heard that all the planets rotate in the same direction(Clockwise) except Venus which rotates in the opposite direction(Counterclockwise). Is this true?
  21. A

    Recognizing a product of two 3d rotations (matrices)

    Hi, I have a problem identifying some 3d rotation matrices. Actually I don't know if the result can be brought on the desired form, however it would make sense from a physics point of view. My two questions are given at the bottom. \mathbf{s}=\left( \begin{array}{c} s_{x} \\ s_{y} \\ s_{z}...
  22. A

    Recognizing a product of two 3d rotations (matrices)

    Hi, I have a problem identifying some 3d rotation matrices. Actually I don't know if the result can be brought on the desired form, however it would make sense from a physics point of view. My two questions are given at the bottom. \mathbf{s}=\left( \begin{array}{c} s_{x} \\ s_{y} \\ s_{z}...
  23. F

    Are Linear and Tangential Acceleration Just Angular Values Times the Radius?

    Question: A fighter pilot is being trained in a centrifuge of radius 15m. it rotates according to theta=0.25(t^3) + ln(t+1) befor it stabilises (theta in radians). what are the magnitudes of the pilots: a) angular velocity: d(theta)/dt at t = 4 (12.2 rad/sec) b) linear velocity: 12.2 *...
  24. U

    Reaching all of R^N by rotations from a linear subspace

    Hi all, I have a problem related to quantum mechanical description of vibrational motion in molecules. I would like, for efficiency, to integrate over symmetries (global rotations) of the molecule. I would like to prove (or disprove) that all points in R^N can be reached by rotations from a...
  25. P

    What is the relationship between SO(n) and S^n?

    Homework Statement Group of rotations of S^1 = SO(2)=S^1 conincidently Group of rotations of S^2 = SO(3) Group of rotations of S^3 = SO(4) Correct? The Attempt at a Solution SO(3) is the group of all rotations in R^3 so it can rotate all elements of S^2 which is part of R^3. Although I can't...
  26. G

    Subatomic rotations in a plane Abelian group

    Hi.. I recently stumbled across a question that seemed a little bit odd "Show that the set of rotations in a plane form a SO(2) Abelian group." for a subatomic physics course. I know how to obtain the answer showing that A^TA=AA^T=1... what I don't understand is what the relevance to subatomic...
  27. G

    Subatomic rotations in a plane Abelian group

    Homework Statement 5. The Z boson has a width of 2.4952 GeV: a. The Z decays 3.363% of the time in e+e-calculate the partial width of Z \rightarrow e+e-. b. The J/ \psi (A cc bar state) has a width of 93.4 KeV. Is its lifetime is longer or shorter than the Z lifetime? Explain. c...
  28. E

    Understanding Divergence Transformations in 2D Rotations

    divergence question show that the divergence transforms as a vector under 2D rotations. I am so confused abouth what this question wants me to do. Obviously the divergence is not invariant under rotations. Consider the divergence of the function f(x,y) = x^2 * x-hat. The divergence is...
  29. N

    Exploring Rotation Matrices: Finite & Infinitesimal Rotations

    Homework Statement Can anyone help me to proceed with this? If we execute rotations of 90* about x-axis and 90* about y axis-what is the resulatant rotation matrix?Will the result commute if we rotate by changing the order?Will they commute if infinitesimal rotations are considered...
  30. O

    How Do Multiple Circle Rotations Affect Trigonometric Curves?

    I have a project in Pre-calculus that I'm not really sure how to do. Our class recently attended Physics Fun Day at FunTown USA. We were each assigned a ride and a set of questions associated with the question. We had a ride called the casino. We figured out how the ride works, and answered most...
  31. R

    Linear Transformation Matrix for Rotations about y-axis

    Derive the matrix for the transformation that rotates a point (x,y,z) counterclockwise about the y-axis through an angle (X). My book gives me a matrice for the y-axis move. (cosX 0 sinX) (0 1 0 ) (-sinX 0 cosX) call the above matrix [A] Im also given this...
  32. A

    Basic question about Pauli Rotations

    So it's apparently possible to prove that e^{-iAx} = cos(x)I + isin(x)A given that A^2=I. What I don't understand is how this is supposed to be derived. Any help would be appreciated as this is driving me nuts and this is probably something that is very easy to prove...
  33. L

    Can space-time rotations create time machines in General Relativity?

    2 Weeks ago..i was reading the novel "The time Ships"... by Stephen Baxter (the second part of "the time machine" by H.G Wells) in a paragraph the "time traveler" explained how his machine worked... inducing a "rotation" in the space time so the "time interval" became an "space interval"..of...
  34. pellman

    Dirac Lagrangian not invariant under rotations?

    First, I need to be able to do equations in my post but it has been a long time since I posted here. Someone please point me to a resource that gives the how-to. If you make a infinitesimal rotation of the free-field Lagrangian for the Dirac equation, you get an extra term because the Dirac...
  35. M

    Rotations, Speed and Directions by eye (may be biology)

    I have a problem thinking about rotations. As we all know, it takes 360 degrees to complete a circle, 720 to do two rotations, 1080 for three full rotations, 1440 degrees for four and 1800 for five and so on. My question is, what is the amount of complete rotations a human eye can see in one...
  36. Oxymoron

    Help with Rotations | ODEs in \mathbb{R}^3

    DISCLAIMER: I would like to post a few things about what I am studying. Hopefully by writing all this I will get a better idea of it. There will probably be a few mistakes. Please feel free to comment on anything I said or add your own point of view, I'd love to hear it! I have three vector...
  37. P

    How Does Quaternion Rotation Affect Another Quaternion?

    If we take a vector "v" and utilize a quaternion q and its conjugate complex, we can rotate the "v" vector this way: qvq* The question is, what happens if "v" is not a vector, and is a quaternion? rotates it?
  38. haushofer

    Rotations in special relativity

    Hi, I have something which bothers me some time and I hope some-one can relieve me from this burden. If one takes the space-time interval in Cartesian Coordinates, one gets ds2=dt2-dx2-dy2-dz2. Ofcourse we could write this in polarcoordinates etcetera. Now, if we want to describe a...
  39. F

    Rotations in Special Relativity

    Greetings--I think I've confused myself about rotational motion in special relativity. Suppose you had a cog-shaped object in space that you caused to rotate by shining a focused beam of light onto its side. Classically, if we treated the light as discrete photons carrying some momentum, the...
  40. P

    Linear transformations and rotations

    Linear transformations and rotations... Hi everyone. I need some help getting started on this question. Let R: R3 ---> R3 be a rotation of pi/4 around the axix in R3. Find the matrix [R]E that defines the linear transformation R in the standard basis E={e1, e2, e3} of R3. Find R(1,2,1)...
  41. K

    Numeric precision with iterative matrix rotations

    I couldn't find a forum section on numerical analysis, so I'm writing this here. I'm on the lookout for simple matrix rotation/multiplication methods that can overcome the precision problems associated with poorly conditioned matrices. In my case I'm trying to simulate the rotational...
  42. V

    Building a Splay Tree: AVL, Single & Double Rotations

    Are there any relations between these two trees ?? Where can i find source code to build a splay tree ? I have heard that if we applied the ideas of SINGLE ROTATION and DOUBLE ROTATION which are used in AVL tree, we can then build SPLAY tree easier, is this correct ? Thanks a lot for any...
  43. marcus

    The rotations, SO(3), in Loop Quantum Gravity

    this is a collective effort to hit the easy parts of LQG for anyone who has shown an interest in this approach to quantum gravity to add constructive input Lubos Motl has been a critic of LQG and has just published a clear presentation of it, plus made an important contribution to and he...
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