Questions tagged [rotation]
Circular motion about a central point or axis
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Equivalence notation Euler angles-angular momentum in Wigner $D$ matrix
In Wigner little-$d$ function the convention that I found in wikipedia https://en.wikipedia.org/wiki/Wigner_D-matrix is z-y-z as shown here.
A 3-dimensional rotation operator can be written as
$$R(\...
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According to inertial frame, how can a bead move in a groove made on a rotating table? [duplicate]
Context:
Consider a smooth circular table rotating uniformly. Along it's radius , a groove is made. While it's rotating , a bead is placed on the groove gently at some distance (say $x$) from centre. ...
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Time Dilation and Spinning Reference Frames
If a spaceship approaches a rapidly spinning planet, would the planet's inhabitants , the inhabitants of the planet where the spaceship came from , and the spaceship's occupants observe time dilation ...
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Topological proof of spin-statistics theorem confusion
I am currently studying the spin-statistics theorem. I have found a section on John Baez's website which presents a "proof" of the spin-statistics theorem. He states the theorem as:
This is ...
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Rotating Reference Frames And Their Phenomenon
In a rotating reference frame, while observing the proper motion of stars due to your spin, would you perceive time dilation when closely observing those stars?
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i've been told that a motor bike can turn on a frictionless horizontal surface by leaning towards the horizontal
their reasoning was that when the bike lean towards horizontal the reaction acting on the bike by surface will act in an angle through the center of gravity of the slanted bike(not perpendicular to ...
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Why can we consider this Hilbert space transformation as corresponding to this rotation of reference frame?
At the outset, let me state what I know. From Chapter 3 of Ballentine (restricting just to rotation transformations since that's all I'm concerned with here) I know that if frame $\mathcal{S}'$ is ...
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Rotation of spin orbit coupling matrix
In first-principle calculation, we could get different solutions for different directions of magnetization for ferromagnetic system.
I want to know the details about whether or when does the SOC ...
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Lattice symmetry operations in strongly spin-orbit coupled systems
I think this is a FAQ when we are studying the rotation operations of lattice spin systems, but I can't find much references.
Background
Considering a Hamiltonian defined on a triangular lattice:
\...
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Do matrices with this property appear in physics?
First I should mention that my background is in Mathematics, but I am looking for a motivating example in physics. I apologize in advance if my question does not meet the standards of this site.
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What is the meaning of the parameters in the axis-angle exponential representation of $\rm SO(3)$?
In the axis-angle parametrization, an element of the rotation group $\rm SO(3)$ is written as
$$R_{\hat{n}}(\theta)=\exp\left[-i(\vec{J}\cdot\hat{n})\theta\right]$$
where $\theta$ represents the angle ...
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Can someone identify this vector representation of $\rm SO(3)$ in terms of multi-variable polynomials?
I am trying to get some deeper intuition for the representations of $\rm SO(3)$ and how they combine with each other, and I ran into an odd object that I'm hoping that folks here might help me ...
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Foucault's measurements of the speed of light
I understand that you can — in principle — measure the speed of light with the rotating mirror experiment.
What I don't understand: How can you accurately measure (or fix) the number of rotations of ...
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When a creaky door briefly makes no sound, has the sound simply gone into the ultrasonic pitch range?
Sometimes, when I’m opening a creaky door, it will only creak for part of the motion; being, apparently, silent for small portions of the turn. However, I’ve noticed that the pitch of the creaking ...
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Proving that rotations of the angular momentum eigenstates are corresponding eigenstates
If I apply a rotation operator about an arbitrary axis to a typical $\mathbf{J},J_z$ angular momentum eigenstate $|j,m \rangle$ then my sense from the development in Ballentine is that I also obtain ...
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How can we show that spin is the generator of rotation? [duplicate]
Starting with the behaviour of wavefunctions, $\psi(\vec{r})$ under rotation,
$$\psi'\left(\vec{r}\right)=\psi\left(R^{-1}{\vec r}\right)$$ it is possible to show that the orbital angular momentum $$\...
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Is it possible to determine a final orientation from an initial angular velocity and constant angular acceleration analytically?
I am looking to model the rotation of a ball over time. I have the following information:
an initial orientation, as a quaternion
an initial angular velocity, as X/Y/Z components, fixed to the global ...
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How to compute linear acceleration in 3D from change in roll, pitch and yaw angles?
We know that if a body is rotating only about $z$-axis along a circle of radius $R$ with an angular rate of $\omega$, then the acceleration of the body in 3D is $a = [0.0\ \ \omega^2R \ \ 0.0]$. Now ...
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What is meant by "rotation group"?
What do physicists mean by the term "rotation group"? Is it synonymous with $SO(3)$? Is it synonymous with $SU(2)$?
I am confused because rotations in real 3D Euclidean space can also be ...
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Maximising spin on a table tennis ball: speed or acceleration?
In table tennis it is often desirable to produce as much spin on the ball as possible using a glancing contact. (The rubber covering of a table tennis bat is typically highly elastic and has a high ...
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Why do we measure plane angle in radians and solid angle in radians and steradians respectively rather than degrees? [duplicate]
Recently, I learnt about physical quantities. When i got to know about plane angle and solid angle, i had a doubt that even though they are just angles, why do we measure it in radians or steradians ...
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How can I optimally rotate accelerometer readings so that the integrated velocity ends up correct? [closed]
I attached an accelerometer (with gyro/magnometer) to a curling rock and threw it down the sheet of ice. The accelerometer was not flat, and it did not travel significantly in the $z$ direction. I ...
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How do we properly describe angular displacements, angular velocities, and the relationship between them (in the most general case)?
If we are merely describing rotations through a fixed axis through the origin, then it is enough to characterize angular displacements by an angle $\theta\in(-\pi, \pi]$. Real-life rotations are not ...
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How is Power Distributed in This Case?
I have been reading the working of induction motor, where I came across Torque-slip characteristics where the equation of torque was determined to be (refer figure).
In the derivation the power lost ...
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Directly get the rotation meaning of $e^{-i\frac{\theta}{2}\sigma _{\vec{n}}}$ from the commutation relation? [duplicate]
Suppose I have three hermitian operators $\sigma _x,\sigma _y,\sigma _z$ that I don't explicitly say they are Pauli matrices but still use the similar notation $\sigma_j$. They only need to satisfy ...
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Why motional emf is induced when a one dimensional object moves?
Suppose we have a conducting rod hinged about a certain point in uniform magnetic field . When we start to rotate the rod with constant angular velocity ω, there will an induced emf of
e = BωL2/2
...
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Apparent paradox in taking torque about a translating axis
Consider a horizontal long rod that is undergoing free fall. Consider the torque about an axis through the rod (perpendicular to the rod and to the direction of gravitational force), that is a little ...
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Why can the dot product of two vectors be expressed as a differential?
I am reading a book by Arfken and Weber (Mathematical methods for physicists), in the section regarding rotations in $\mathbb{R}^3$. They express the elements of a rotation matrix in Cartesian ...
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Unexplainable discrepancy between the centripetal force calculated directly and by linear regression
The centripetal force $F_C$ of a uniform circular motion can be expressed as,
$$F_C=\frac{4\pi^2mr}{T^2}$$
where $m$ is mass, $r$ is the radius, and $T$ is the time interval for one revolution (the ...
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Angular momentum about an axis?
We know that angular momentum of a body is defined about a point in space. Let us consider a solid cylinder whose radius is R and mass is M. It has a moment of inertia defined around the axis of ...
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Can an object have more than one axes of rotation? [duplicate]
A few answers I found say "no." Perhaps because the conditions for rotation around multiple axes have not been met. However, I have seen a couple of videos of objects spinning around both ...
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Why dimension units of radius is not $\rm m/rad$ or $\rm cm/rad$? [duplicate]
Radius is not a just simple size or length between the two points. The radius shows the connection of linear and angular values. Something must indicate the information about a perpendicularity of the ...
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Rotate an object about the time axis
Is there a notion of rotating an object about its time axis? I'm not sure if this question totally makes sense, but it seems intuitive to me that an object with dimensions in the three spatial ...
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What is the derivative of general 3D rotation with respect to one angular component? [closed]
For a general rotation $R(t_1, t_2, t_3)$ where the $t_i$'s are the components of the rotation vector in the axis-angle representation. Is there closed formula for the derivative of $dR/dt_i$?
I only ...
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Measuring the effect of spin of a tennis ball on its trajectory
Upward spin (lift) applied to a tennis ball will shorten its trajectory.
Are mathematical calculations and actual experimental results on this available somewhere?
If not, does anyone know how to ...
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What's the difference between spherical symmetry and rotational symmetry in quantum mechanics?
I am reading quantum mechanics and these two concepts are confusing me.
Griffths' QM book says "perturbation $H'\sim p^4$ is considered "spheric symmetric", so it commutes with $L^2$ ...
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How to find Moment of Inertia at a different axis?
So I have a solid disk:
m = 2.98 kg
r = 0.2 m
and an axis in the - and + z direction (in unit vector form, the k dimension)
And I have to find the Moment of Inertia of the disk if the axis is at point ...
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Why is operator $p^2$ rotationally invariant in QM? [closed]
It seems a trivial question but I don't know something explicitly.
We suppose $\hat{H}=\frac{\hat{p}^2}{2 m}$ and we have a rotation defined by $\hat{R}_{\mathbf{n}}(\varphi)=\exp \left[-\frac{i \...
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Computing rotation matrices - non-commuting operators
A rotation matrix parametrized by Euler ZYZ angles, $\alpha, \beta, \gamma$ can be written as:
$$
\hat{R}(\alpha, \beta, \gamma) =
\exp{\left( -i\alpha\hat{J}_{z} \right)} \cdot
\exp{\left( -i\beta\...
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Is this a correct theoretical concept of a simple electric current generator?
As I think I somehow understand electromagnetic induction and after watching several experiments on YouTube with magnet pieces left to fall through coils which had connected ends to permit the ...
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Why is $\sum_{i}m_iv_i$ equal to zero for a rolling ball?
I was watching a video about rotational kinetic energy in which they derived the formula $$K=\dfrac{1}{2}mv^2+\dfrac{1}{2}I\omega^2$$
each point on the circle has a translational velocity equal to the ...
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Generators of Angular Momentum
This semester I'm taking Quantum Mechanics II and we are in the theory of angular momentum. One particular thing got me thinking: when one does the representation of the rotations of the $SO(3)$, a ...
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What is the matrix for change of basis from unrotated to an airplane with yaw, pitch and roll?
From the wiki page on rotation matrix:
I find this rather confusing. I understand that the resulting matrix rotates a vector roll around the fixed $x$-axis followed by pitch around the fixed y-axis ...
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How do I convert vectors between rotated cartesian coordinate systems? [closed]
I am trying to understand how to transform vectors between cartesian coordinate systems that are rotated wrt each others. As a step in this process I have come up with a toy problem:
Consider a ground ...
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Why is there a horizontal acceleration if there is no horizontal force?
Assume a non uniform sphere, on a smooth horizontal floor. Its center of mass isn't at its geometrical center, but on the line passing through geometrical center and parallel to horizontal floor.
My ...
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Angular momentum and velocity about a point about which a rigid body is not rotating
Suppose a rigid body of mass $m$ is rotating about its centre of mass with angular velocity $ω$, and the centre of mass is translating with linear velocity v, consider two cases:
(I) we want to ...
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Can a rotating body see it's own reflection?
So my question is this: Say I'm able to get close to, but not achieve the speed of light. I want to look at the back of my head in the mirror, would it be possible to turn around before the light from ...
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Rotations and angular momentum
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 ...
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Kinetic energy in 3d rotational movement
When we calculate the angular momentum in this example we have 3 components (1 orbital and 2 spin components). Does the same hold for its Kinetic Energy? Like this:
$$
K=\frac12\left(I_s \Omega^2 + I_\...
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Goniometer stages/positioners stacking
I am working on a project related to calculation of roll/pitch/yaw misalignment. I am new to the field and trying to understand. I more or less understand math behind this - order of rotations is very ...
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