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.
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...
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.
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...
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...
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...
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...
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...
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...
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...
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...
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...
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.
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...
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...
The doubt is about B and C.
b)
n = 4, $C = {I,e^{2\pi/4}}
n = 5, $C = {I,e^{2\pi/5}}
n = 6, $C = {I,e^{2\pi/6}}
Is this right?
c)
I am not sure what does he wants...
I first computed the operator ##\hat{T}## in the ##a,b,c## basis (assuming ##a = (1 \ 0 \ 0 )^{T} , b = (0 \ 1 \ 0)^{T}## and ##c = (0 \ 0 \ 1)^{T}##) and found
$$ \hat{T} = \begin{pmatrix} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{pmatrix}.$$
The eigenvalues and eigenvectors corresponding to this matrix...
Can we consider the E and B fields as being irreducible representations under the rotations group SO(3) even though they are part of the same (0,2) tensor? Of course under boosts they transform into each other are not irreducible under this action. I would like to know if there is in some...
Good morning!
I have a problem in understanding the steps from vectors to operators.
Imagine you are given a vectorial observable.
In classical mechanics, after rotating the system it transform with a rotation matrix R.
If we go to quantum mechanics, this observable becomes an operator that is...
Hi
I am using Kleppner and it states that finite rotations do not commute but infinitesimal rotations do commute. I follow the logic in the book but i don't understand the concept. Surely a finite rotation consists of many , many infinitesimal rotations and if they commute why doesn't the finite...
The calculations for the magnetic field produced by a uniformly rotating charged sphere can be found in basically every book on electrodynamics. I wonder what happen with the magnetic fields produced by rotating rigid solid that also present precession and nutation movements.
The question comes...
Hey! :o
Let $ \tau_v: \mathbb{R}^2 \rightarrow \mathbb{R}^2, \ \ x \mapsto x + v $ be the shift by the vector $ v \in \mathbb{R}^2 $.
Let $ \sigma_a: \mathbb{R}^2 \rightarrow \mathbb{R}^2 $ be the reflection on the straight line through the origin, where $ a $ describes the angle between the...
Hi,
I was looking at the so(3,3) Lie algebra which has 3 temporal rotation generators as well as the normal 3 spatial rotation generators. When I try to use Noether's Theorem to determine what the conserved quantity is, due to invariance under temporal rotations, I seem to get an integral where...
Suppose I have a positive spin-##1/2## eigenstate pointing in the ##z##-direction. If I apply a rotation operator by an angle ##\theta## around the ##z##-axis the state should of course not change. However, if I write it out explicitly, I find something different:
$$R_z(\theta)|\uparrow\rangle =...
A gear A has 50 teeth and another B has 10 teeth, how many times does the small gear rotate around the big one? I thought 5 but its 6! Note: The gear is curved like 360 degrees.
In 3D the most general motion of a rigid body consists of a displacement and a rotation.
In higher dimensions is this still the most general motion? Or are there unexpected ways of moving with more freedom?
One subtlety, for example, is that we would have to allow for multiple rotations...
Here's a question I want to ask a physicist:
Is it possible for a hurricane in Earth's northern hemisphere to have sufficient linear momentum (directed South) to cross Earth's equator and still persist as a CCW-rotating hurricance but in the southern hemisphere? If so, for approximately how much...
I'm tasked with drawing the trajectory of the Moon around the Earth (in 2D), taking into account the fact that the trajectory also undergoes precession, so the elliptical orbit rotates about it's center (I think it should rotate around the Earth-Moon barycenter, but for the first step I...
Summary: Rotate a shape back to it's original position with the least amount of rotations.
Lets say you have a cube. It's starting rotation is (0,0,0).
It can be rotated on each axis ( x,y,z ) no more than once each by 90 or -90 degrees
(rotation can also be skipped for any axis).
The shape...
Hi
I have used cross products thousands of time without really knowing what it actually does; I know how to compute it, but I don't feel like I understand it. Also, when it shows up in physics/kinematics contexts, it's only because the magnitudes of the vectors involved have to be multiplied...
The system is in rotational equilibrium and therefore experiences no net torque, meaning all individual torques must add to zero.
τNET = 0 = FFTsin(θ)L - FgL - Fg(L/2)
τNET = 0 = FTsin30°(0.6?) - (0.5)(9.8)(0.6) - (2.0)(0.6/2)
My only problem (I think) is figuring out what the length L is for...
I am trying to understand the picture below which is of a contractible and uncontractible loop in what I would call (proper name?) "rotation space", where "rotation space" is a solid ball of radius π with opposite points on the surface of the ball identified, each point of the ball representing...
Looking into the infinitesimal view of rotations from Lie, I noticed that the vector cross product can be written in terms of the generators of the rotation group SO(3). For example:
$$\vec{\mathbf{A}} \times \vec{\mathbf{B}} = (A^T \cdot J_x \cdot B) \>\> \hat{i} + (A^T \cdot J_y \cdot B)...
Being scratching my head for 2 days and not getting anywhere with this one. I am trying to figure out how to perform a 3D rotation described via a mix of intrinsic and extrinsic angles.
Here is the problem:
I have a platform in the shape of a box with sides of length lx, ly and lz. The platform...
I'm trying to wrap my head around the concept. we use 3 rotations to transfer our regular cartesian coordinates (3 x,y,z unit vectors) to other 3 unit vectors. each rotation is associated with an angle. so far I'm good.
but now I saw in Landau's and Lifshitz's "mechanics" book this thing...
I am trying to figure out how to solve this equation. I have a car with tires of diameter 28", and they rotate 10,000 times. How far did I travel?
According to my textbook it's 13.9 miles.
I can figure it out by finding the circumference of the tire (87.96"), multiplying that by 10,000...
I currently styding applications of Lie groups and algebras in quantum mechanics.
U^{\dagger}(R)V_{\alpha}U(R)=\sum_{\beta}R_{\alpha \beta}V_{\beta}
Where ##U(R)## represents rotation. Letter ##U## is used because it is unitary transformation and ##R_{\alpha \beta}## matrix elements of matrix...
One of the reasons I've been so stumped about learning about angular momentum in QM, is that in my classical physics class we only applied it to circular motions. Hence, while I am aware that angular momentum is connected to spherical harmonics, the orbital shapes (besides s) isn't really...
In the momentum representation, the position operator acts on the wavefunction as
1) ##X_i = i\frac{\partial}{\partial p_i}##
Now we want under rotations $U(R)$ the position operator to transform as
##U(R)^{-1}\mathbf{X}U(R) = R\mathbf{X}##
How does one show that the position operator as...
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...
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...
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...
< 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.
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...
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...
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...
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*...
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...
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}...