What is Rotation: Definition and 1000 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

    Matrix of eigenvectors, relation to rotation matrix

    So I am given B=\begin{array}{cc} 3 & 5 \\ 5 & 3 \end{array}. I find the eigenvalues and eigenvectors: 8, -2, and (1, 1), (1, -1), respectively. I am then told to form the matrix of normalised eigenvectors, S, and I do, then to find S^{-1}BS, which, with S = \frac{1}{\sqrt{2}}\begin{array}{cc} 1...
  2. Petrus

    MHB Surface area of rotation about the y-axis

    Calculate the area of ​​the surface of rotation which occurs when the curve rotate in y-axe. I start with x=\sqrt{28y} then f'(x)=\frac{14}{\sqrt{28y}} so we got 2\pi\int_0^{21}\sqrt{28y}\sqrt{1+(\frac{14}{\sqrt{28y}})}^2 then I rewrite as \int_0^2\sqrt{28y}\sqrt{1+\frac{196}{28y}}...
  3. R

    Spinor representation of rotation

    Homework Statement Show the spinor representation corresponding to the rotation through an angle θ about an axis with direction vector n = (n_x, n_y, n_z) has the form: g=exp{-i\frac{θ}{2}(n_x σ_x+n_yσ_y+n_zσ_z)}, σ_{x, y, z} are respectively Pauli matrixHomework Equations h=gxg^{-1}The...
  4. C

    Free Pivot Rotation: Calculating Kinetic Energy, Angular Speed & Linear Speed

    A thin, cylindrical rod = 26.6 cm long with a mass m = 1.20 kg has a ball of diameter d = 10.00 cm and mass M = 2.00 kg attached to one end. The arrangement is originally vertical and stationary, with the ball at the top as shown in the figure below. The combination is free to pivot about the...
  5. R

    Trouble with Wick rotation in 1+1d abelian Higgs model

    When solving for instanton solutions in a 1+1d abelian Higgs model, it's convenient to work in Euclidean space using the substitution x^0 \rightarrow -ix_4^E,\quad x^1 \rightarrow x_1^E The corresponding substitution for the covariant derivative is D^0 \rightarrow iD_4^E,\quad D^1 \rightarrow...
  6. P

    Why dont we feel earths rotation?

    Earth is completing one rotation in one day i.e, in 24 hours. Its equator length is around 40,000 km. Every point on equator is moving with a speed of 40,000/24 i.e, 1667 km/hr. Why don't we feel that speed even though we are moving at such high speeds when we are at equator?
  7. A

    Rotation without slipping and the direction of friction?

    sorry, this is is a general question about a conceptual definition I read in my textbook, i hope that's ok. "an object that rolls without slipping at a constant velocity over a surface with friction experiences no frictional force" is this true? i understand that on a frictionless surface, the...
  8. P

    Doubts on Rotation: Seeking Physical Understanding

    I have some doubts on topics related to rotation, so I thought I'd make a single thread rather than multiple ones. 1. Why are the Coriolis and Centrifugal forces "fictitious"? I think I might be getting confused by the terminology, but do they represent physical forces? My lecture notes say...
  9. A

    Ethane rotation at room temperature

    The energy available at room temperature is 0.593 kcal/mol (wikipedia) so why is it that Ethane is said to freely rotate from staggered to eclipse if it has a rotational energy barrier of 2.9 kcal/mol (wikipedia)? What am I missing here?
  10. K

    Rotation Problem: Angular Displacement & Spin Time Calculation

    Homework Statement A coin is spinning on its edge at 5 rotations per second. Friction slows down its spin rate at .4 r/s2 a) what angular displacement does the coin have by the time it's slowed down to half its original angular velocity. b) how long before the coin stops spinning?
  11. K

    Merry go round Rotation Problem

    Homework Statement A merry - go - round rotates clockwise 15 times in 45 seconds. By rubbing against its edge, a child stop it from turning in 15 seconds. a) find its initial angular velocity (ω) b) find its angular acceleration (\alpha) The Attempt at a Solution 15 times in 45...
  12. Z

    Rotation of a uniform rigid disc about a fixed smooth axis

    Homework Statement A uniform circular disc has mass M and diameter AB of length 4a. The disc rotates in a vertical plane about a fixed smooth axis perpendicular to the disc through the point D of AB where AD=a. The disc is released from rest with AB horizontal. (See attached diagram) (a)...
  13. V

    Calculate the equatorial rotation velocity

    (If this is in the wrong section, please feel free to move it.) Hi all, How would one calculate the equatorial rotation velocity or rotation period without the other? Is this possible using the values; diameter, circumference, mass and revolutionary/orbital period? Thank you.
  14. Y

    Rotation matrix arbitrary axis?

    Hello everyone. I'm having some trouble with rotation matrices. I'm given three matrices J_1, J_2, J_3 which form a basis for the set of skew-symmetric matrices (\mathfrak{so}_3). Further, the matrix exponent function is such that exp[\alpha J_i]=R_i(\alpha), so taking the...
  15. S

    Question on hyperbolic rotation

    Hello, I see that hyperbolic rotation of frame F' about the (x2,x3)-plane of frame F is identical to a Lorentz transformation, corresponding to a linear motion along x1 of the frame F' with respect to F. Then hyperbolic rotation about (x1,x2) means motion along x3 and hyperbolic...
  16. X

    How Does a Ball's Velocity and Acceleration Change on a Circular Path?

    Homework Statement A ball moving in the circular path with a constant speed of 3.0 m/s changes direction by 40.0 degrees in 1.75 seconds. What is the change in velocity? What is the acceleration during the time? Homework Equations Fc = m * ac ac = v^2/r The Attempt at a Solution...
  17. B

    Torque, point or axis of rotation

    Hi, 1) Do we calculate torque with respect to a point or with respect to an axis? I have read them both in different resources, and so I am confused! 2) If we calculate torque with respect to an axis, many introductory textbooks discuss the motion of the gyroscope by considering how the...
  18. D

    Max Rotation Rate of a Slowing Turntable

    Homework Statement A turntable of radius "r" is spinning counterclockwise at an initial rate of ω. at t=0, its rotation rate begins to slow at a steady pace. the rotation finally stops at t=T. At what time is ωt^2 maximum. Express your answer in terms of T.
  19. shounakbhatta

    Rotation speed of a neutron star

    Hello, Can somebody please tell me in details about the rotation speed of a neutron star? Does it rotate very fast and then slows down? Thanks.
  20. A

    Rotation of Gridded Spherical Coordinates to the Same Grid

    I have a uniform grid of data in spherical coordinates. e.g. theta = 0, 1, 2, ... 180 and phi = 0, 1, 2, ... 359 which forms a 2D matrix. I wish to rotate these points around a cartesian axis (x, y, z-axis) by some angle alpha. To accomplish this I currently do the following: 1. Convert to...
  21. R

    Axis of Rotation - Find Solution

    Homework Statement consider the following rotation matrix: 0 0 1 1 0 0 0 1 0 Find the axis of rotation. Homework Equations The Attempt at a Solution I know the following: Ω|1> = |2> Ω|2> = |3> Ω|3> = |1> where Ω is an operator. It is a cyclic permutation. What do not understand is...
  22. A

    Does rotation of bullet when fired increases its lethal power?

    Hi! I was watching Discovery channel the other day and it was showing a series on evolution of guns and ammo.. It showed that earlier there was bolt mechanism for firing the gun. And then, a revolution came when muzzzle or bullet was shot such that it went in forward direction while...
  23. P

    Wick rotation and Minkowski/Euclidean space

    Hello, I was wondering why in integrals such as \int d^4k F(k^2) where k^2 = (k^0)^2 - |\vec{k}|^2 ranges from -∞ to ∞, once the Wick-rotation is performed, we have -k^2_E = -(k^0_E)^2 - |\vec{k}_E|^2 which lies in the (-∞,0) interval ... So, the contribution which lies in the part...
  24. D

    MHB Angular velocity and rotation

    The telescopic boom of a crane rotates with the angular velocity and rotation as indicated about point $A$. At the same instant, the boom is extending with a constant speed of 0.5ft/s, measured relative to the boom. Determine the magnitude and acceleration of the absolute acceleration of point...
  25. J

    Partial derivatives of 3D rotation vectors

    I am utilitizing rotation vectors (or SORA rotations if you care to call them that) as a means of splitting 3D rotations into three scalar orthogonal variables which are impervious to gimbal lock. (see SO(3)) These variables are exposed to a least-squares optimization algorithm which...
  26. G

    Using two positions on a rigid body to calculate rotation

    Hello all, I'm currently a undergrad university student doing research and I'm anaylising some position data. The data is a time-series' of Eastings (x) and Northings (y) for two points (P1, P2) on a rigid body in motion (T, E1, N1, E2, N2), with a position reported every 1 minute. I know...
  27. A

    Exploring the Mathematics of Rotation: A Mathematician's Perspective

    For a mathematician, how is rotation defined in the most general sense? Question arose to me because it occurred to me that an essential property of the rotation matrix is that it preserves lengths. Is this the only mapping (if not please give me a counterexample) that has this property...
  28. J

    Change in Earth's axis and Rotation due to tectonic activity

    Since the redistribution of mass in the Earth's surface can be caused by earthquakes, sometimes the Earth's rotation is increased or decreased by a small amount. Recent series of quakes seem to be related. To me, this makes sense, since if plate "A" should move, then plate "B" would also...
  29. P

    How can I use different angles in linear algebra rotations?

    hi, I understand how to do the rotation equation A = [ cosθ -sinθ sinθ cosθ] A*v = [ cosθ -sinθ * [ x = [ xcosθ - ysinθ sinθ cosθ] y ] xsinθ + ycosθ ] A*v = [ cos90 -sin90 * [ 6 sin90 cos90 ] 4 ] =...
  30. M

    Rotation Operator: Spin 1/2 vs Spin 1

    How does finding the rotation operator for a spin 1/2 particle differ from finding that of a spin 1 particle?
  31. Z

    Rotation of Rigid Body: Analyzing Classical Mechanics

    In the framework of Classical Mechanics,there is no problem in the rotation of rigid body.But I want to make the concept about rotation more clear. About rigid body ,there is no vibration;it would only rotate and translate.We can easily distinguish translation from rotation in rigid...
  32. L

    Rotation on a Plane | Using Rotation Matrix and Point Rotation

    Homework Statement On a X-Y plane we have a square with its 4 corners A(3,1) B(7,3) C(2,6) D(0,2). We are to rotate the rest of the square around the point A clockwise by 70 degrees. Homework Equations (I am not sure how they are called in English) The rotation matrix 2x2 1st row...
  33. shounakbhatta

    Stars Rotation | What is Angular Momentum?

    Hello, Do stars rotate? I mean to say that do they rotate in angular momentum? Or there is any other rotation?
  34. T

    Effect of earth's rotation on an object on the surface

    Hi, This is one of the things that confuses me. Assume an object on the surface of the Earth has a mass m and F=m*g is the force on it due to the gravity. But also the Earth is rotating and although the radius is extremely large compared to the size of the object but it must be affected...
  35. I

    Efficiently Solve a Challenging Rotation Problem | Homework with Shell Method

    Homework Statement Homework Equations Shell method The Attempt at a Solution Not sure if this is right, but the integral I set up is: 2\pi \int_0^1 (4-y)(\sqrt{siny})dy Finding the radius is my big problem with these problems, I can't visualize it very well. Any help is...
  36. A

    How does the curl equation measure rotation?

    For a 2D vector field {F}=P(x,y)\vec{i}+Q(x,y)\vec{j} curl {F} = \frac{\partial Q}{\partial x}+\frac{\partial P}{\partial y}\vec{k} So that's the rate of change of the j component of a field vector with respect to x plus the rate of change of the i component with respect to y...how does...
  37. N

    Tether rotation device in space problem

    Homework Statement A spaceborne energy storage device consists of two equal masses connected by a tether and rotating about their center of mass. Additional energy is stored by reeling in the tether; no external forces are applied. Initially the device has kinetic energy E and rotates at...
  38. R

    [QM] Total angular momentum rotation operator

    Homework Statement How to prove that for any representation of the spin, the state e^{-i{\pi}J_x/\hbar}|j,m\rangle is proportional to |j,-m\rangle The exponential term is the rotation operator where J_x is the x-component of the total angular momentum operator, and |j,m\rangle is an...
  39. A

    Inverse 2D rotation with negative parameters

    Homework Statement Say we have a 2D rigid body transform, with parameters p = [ p_1,p_2,p_3] for rotation, x translation and y translation respectively. I'm using the transform to .. transform an image. Is there a way to have: For a point x , y : x',y' = transform(x,y,p) <=> x,y =...
  40. S

    Wick's Rotation contour integration

    Hey, I have Wick's rotated a contour integration of the form \frac{i\lambda}{2}\int d^{3}p\int \frac{dE}{(2\pi)^{4}}\frac{1}{E^{2}-p^{2}-m^{2}} this is the form where we integrate along the real line, we rotate this to 'Euclidean' Space such that we make the changes E\rightarrow iE\...
  41. R

    Causing levitating magnet rotation

    I have a device like this one: http://www.youtube.com/watch?v=kA_hw_lY-OY (All of them work the same) Here's an image: I want to make it spin without touching it with hands. I tried using magnets of several kinds, and tried to put them in many places around it but it didn't make it...
  42. T

    Volumes by Slicing and Rotation about an Axis.

    Homework Statement Find the volume of the solid generated by revolving the region between the parabola x = y^2 + 1 and the line x = 3 about the line x = 3. Homework Equations The answer is found by integrating with respect to y with disk method, but I don't understand why my answer is...
  43. H

    Understanding the Contradiction: Spin Rotation in Quantum Systems

    This has been a contradiction in my brain for some time. If I want to rotate one nuclei (spin 1/2), with say an applied magnetic field B and RF pulse (at the appropriate larmor frequency), how does the spin actually rotate? I thought it can only take on discrete values of 1/2 or -1/2...
  44. C

    How do I calculate the new coordinates after rotating the coordinate axes?

    If I have a point (x,y) and I rotate the axises by some amount. Why is x' = xcosθ+ysinθ and y'=-xsinθ+ycosθ?
  45. H

    Question about rotation w/ an additioanl object.

    First, while this is "homework" related I'm not seeking any direct answers. I'm stuck on a concept that I'm not sure even exists. In short I've got a rotating disk with a weight at the edge with "an initial" ω. If that object was to decrease the distance towards the center, would the ω...
  46. P

    Questions - solving pendulum period of rotation

    Questions -- solving pendulum period of rotation Physical pendulum has a period of rotation that changes by Changing the temperature. According to the above show that the frequency shift of the pendulum would be: I have 2 questions,first solving the above question and the second how...
  47. H

    How to use Euler's angle theorem in rotation of a coordinate

    If i have a point at (0,0,5) in x,y,z system, then i make 2 rotation on the point with center at origin. i)the first rotation is on y-axis with angle P in clockwise direction. ii) the second rotation is on the point's new x-axis rotate in angle Q in clockwise direction. How can i find the...
  48. B

    Rotation Problem Involving Two Disks

    Homework Statement I attached the problem as a word file. Homework Equations The Attempt at a Solution I think I will be able to solve this problem; my only questions is, can I treat the two disks as one, by summing their rotational inertias together?
  49. S

    Does rotation of a 3d volume about a 2d plane create gravity?

    Hello, If a rotation of a 2-d disk about a 1-d line can produce centrifugal gravity, is it right to infer that rotation of a 3-d volume about a 2-d plane would create gravity, observable in the 3-d volume? It is confusing to think about the direction of this gravity, if it exists...
  50. A

    Rotation and Revolution in relativity

    I have a question about the concepts of rotation and revolution - on how they are treated in relativity. Since all motion is relative, a revolution of a planetary body around a central body could also be seen instead as a rotation of the central body w.r.t. a fixed (non-revolving) planetary...
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