Rotation Definition and 1000 Threads

Hello! I was reading two things: 1) tidal locking (as explained in the Wikipedia article:https://en.wikipedia.org/wiki/Tidal_locking where it is stated that, because of internal friction caused by the body of water being attracted to the moon and deforming, the kinetic energy of the system...

Hi, I started with calculating the moment of inetria of the rod: I = ⅓ML^2 + M(3/2 * L)^2 = 31/12 ML^2 and I thought that the reaction force in the first case will be equal to centrifugal force: F1 = Mω^2*(3/2)L Angular velocity is calculated from the conservation of energy: Mg3/2*L=1/2 * Iω^2...

Is any way to get Rodrigues' rotation formula from matrix exponential \begin{equation} e^{i\phi (\star\vec{n}) } = e^{i\phi (\vec{n}\cdot\hat{\vec{S}}) } = \hat{I} + (\star\vec{n})\sin\phi + (\star\vec{n})^2( 1 - \cos\phi ). \end{equation} using SO(3) groups comutators properties ONLY...

I started by inserting ##ds=\sqrt{dx'^{\mu} dx'_{\mu}}## and ##p'^{\mu}=mc \frac{dx'^{\mu}}{ds}##. So we have: $$\frac{dp'^{\mu}}{ds}=mc \frac{d}{dx'^{\mu}} \frac{d}{dx'_{\mu}} (x'^{\mu})$$ Now I know that ##dx'^{\mu}=C_\beta \ ^\mu dx^\beta## and ##dx'_{\mu}=C^\gamma \ _\mu dx_\gamma## where...

the device that I have is the same as the mirror of this truck.

This comes from a list of exercises, and setting ##m_1 = 5.4kg##, ##m_2 = 9.3kg## and ##F=5N##, the answer should yield ##2.19m/s^2## (of course, supposing the answer is right). If I knew the radius ##R## of the cylinder, I could find its momentum and use it to find the linear acceleration...

Hello there, I have a question regarding this problem. I have no problem with part A. However, in part B, my solution manual states that the hollow cylinder will reach the bottom last. Why is it? I mean shouldn't the solid cylinder and the hollow one reach the bottom at the same time? you know...

First I found the moment of inertia, I=1.8(5.5^2+3.9^2+4.9^2) =125.046 Then I tried to find the rotation rate using the equation L=rotation rate*I rotation rate=3773/125.046=30.173 But the answer is suppose to be 21.263?

Hi, I have the following problem: A homogeneous disc with M = 1.78 kg and R = 0.547 m is lying down at rest on a perfectly polished surface. The disc is kept in place by an axis O although it can turn freely around it. A particle with m = 0.311 kg and v = 103 m/s, normal to the disc's surface at...

In high school I learned about three kinds of motion in classical mechanics - translation, rotation, and oscillation. Are there any other kinds of motion in the physical world?

Is third Newton law valid for rotation / Torque? I mean, can we say that for every torque there must be another torque with equal magnitude and opposite direction? This can only be true for contact forces or radial forces, as these forces will create a reaction that will cancel the torque...

This is one of very few in-depth sources of information I can find online about Highly Coupled Magnetic Resonance...

This one is challenging for me. Can anyone give me a hint?

Hi. I don't know what prefix this question belongs in so I just chose advanced at random. What's the physical effect called when the Earth orbits around the sun at extremely fast speeds and also rotates around itself every 24 hours at the same time? Does that force cause anything in space...

Let's begin with the first point. a.I) Apply a generic boost in the y-z plane (take advantage of the arbitrariness in deciding the alignment of the y and z axes). \begin{equation*} B_{yz} = \begin{pmatrix} \gamma & 0 & -\gamma v_y & -\gamma v_z \\ 0 & 1 & 0 & 0 \\ -\gamma v_y & 0 &...

So for the first one on the left is it T= 2g*20=40Nm ? Our professor said it is clockwise I just don't understand how she gets that conclusion.

Please look at this YDSE with two orthogonal polarizers...

Hi, I take a big number of disks to composed a circle of a radius of 1 m, the blue curved line is in fact several very small disks: I take a big number of disks to simplify the calculations, and I take the size of the disks very small in comparison of the radius of the circle. The center A1 of...

Ok, I know there are a lot of strange things in our solar system. Can anyone explain why the small planets spin so slowly? and why does Jupiter spin so quickly? It seems like a ball of debris, getting smaller and smaller, would increase its speed like an ice-skater pulling their arms in...

Hi all:) In my recent exploration of Elliptic Function, Curves and Motion I have come upon a handy equation for creating orbital motion. Essentially I construct a trigonometric function and use the max distance to foci as the boundary for my motion on the x-plane. When I plot a point rotating...

In Rigid body rotation, we need only 3 parameters to make a body rotate in any orientation. So to define a rotation matrix in 3d space we only need 3 parameters and we must have 6 constraint equation (6+3=9 no of elements in rotation matrix) My doubt is if orthogonality conditions...

So when the rotation starts some water will move upwards and in the vertical part of tube. I know hat centripetal force will be given by F=mv²/r Now I though of taking r as centre of mass of the water system but I don't know what to take the value of m as? Should I only consider the water...

From a freebody analysis I got, $$ \vec{r} \times \vec{F} = |r| |F| \sin( 90 - \theta) = (R-r) mg \cos \theta$$ and, this is equal to $$ I \alpha_1$$ where the alpha_1 is the angular acceleration of center of mass of small circle around big one, $$ I \alpha = (R-r) mg \cos \theta$$ Now...

The speed of the ends of the galaxies is higher than what it should be. Current solution: This could be explained by hypothetical "dark matter", which was not found up to now, or by a MOND theory (MOdified Newtonian Dynamics). Can this be explained instead with rotational frame-dragging...

I watched a video that showed how to calculate the center of gravity of a horizontal bar suspended from two wires, one attached to each end. Each wire was then attached to a vertical wall. The angle each wire made with the wall it was attached to was given. They treated it as an a example of...

When the lamina rotates about A, FA must act on B (because it is the farthest away) perpendicular to AB (so that all of FA contributes to rotation). Same argument is valid for rotation of lamina about B as well. Having noted that, I tried two approaches: Approach 1- If I assume that the...

I had a question from the magnetic dipole thread that was posted earlier today, but it's a bit more mundane. The torque on a magnetic dipole, using a right handed cross product is ##\vec{\tau} = \vec{\mu} \times \vec{B}##. The work done during a rotation is $$W = \int \vec{F} \cdot d\vec{r} =...

I am afraid I had no credible attempt at solving the problem. My poor attempt was writing the matrix ##\mathbb R## as a ##3 \times 3## square matrix with elements ##a_{ij}## and use the matrix form of the orthogonality relation ##\mathbb R^T \mathbb R = \mathbb I##, where ##\mathbb I## is the...

I have had a thought experiment in my head for a while now and I am unable to find clear enough examples/info that deal with similar issues, to solve it on my own. This is why I hope that someone in this forum can at least point me towards a solution or provide hints as to where should I be...

Let me start with the rotated vector components : ##x'_i = R_{ij} x_j##. The length of the rotated vector squared : ##x'_i x'_i = R_{ij} x_j R_{ik} x_k##. For this (squared) length to be invariant, we must have ##R_{ij} x_j R_{ik} x_k = R_{ij} R_{ik} x_j x_k = x_l x_l##. If the rotation matrix...

Hi :) 1/ First case A wheel with a mass ##m## and a radius ##r## moves in horizontal translation and rotates around itself. The wheel is just above the ground, doesn't touch it. The wheel rotates CW if the wheel moves in translation to the right. The ground is horizontal and there is no...

A single pair of points will be in contact between P and Q. The frictional force will try to make the velocity of these points equal. Say the final angular velocity of Q is ωq. The velocity of points in contact can never be equal because of difference in directions of ωq and ωp. If I break...

mgR = d(mvR + MvR + ½M(R^2)v/R)/dt mgR = ma + Ma + ½Ma mg = a(m + 3/2M) v = mgt / (m+3/2M) My answer is incorrect. The right answer is v = mgt/(3m+M), but I have no idea what I'm doing wrong.

I use an example with a rack and a pinion. I suppose there is no losses from friction. I suppose the masses very low to simplify the study, and there is no acceleration. I suppose the tooth of the pinion and the rack perfect, I mean there is no gap. There is always the contact between the rack...

$\displaystyle\pi\int_{0}^{\infty} e^x \ dx = \pi$ Ok I looked at some of the template equations but came up with this.

If two coordinate systems are related by a rotation or a boost, does it make sense to say the tensors components are rotated or boosted with respect to their components in the original coordinates? For vectors, I think it is standard to say that, but what about general tensors?

[Mentor Note -- OP deleted his posts after receiving help. His posts are restored below] @ocean1234 -- Check your messages. Deleting your post is not allowed here, and is considered cheating. Problem was given: ##\theta(t) = at - bt^2 + ct^4## a) calculate ##\omega(t)## b) calculate...

In summarize, i have four equations and five incognits. T2,T1,theta,a2,f Need to find one more equation, but i don't know how

I was solving a problem for my quantum mechanics homework, and was therefore browsing in the internet for further information. Then I stumbled upon this here: R is the rotation operator, δφ an infinitesimal angle and Ψ is the wave function. I know that it is able to rotate a curve, vector...

Hey! 😊 Let $\delta_a:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be the rotation around the origin with angle $\alpha$ and let $\sigma_{\alpha}:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be the reflection about a line through the origin that has angle $\frac{\alpha}{2}$ with the $x$-axis. Let $v\in V$...

I've had a question bugging me lately and no matter how much I google, I can't seem to find an answer. I'm sure this probably isn't possible, and my logic is probably flawed (Earth sciences isn't my forte), so I figured I'd ask you fine folks here. First, my understanding of wind, it's caused...

The below matrix represents a rotation. 0 0 -1 0 1 0 1 0 0 Im trying to obtain the general point (x y z) when rotated by the above rotation matrix? So visited https://www.andre-gaschler.com/rotationconverter/ entered the above figures and not sure which entry would be x y z but assume it...

Below is the attempted solution of a tutor. However, I do question his solution method. Therefore, I would sincerely appreciate it if anyone could tell me what is going on with the below solution. First off, the rotation of the matrix could be expressed as below: $$G = \begin{pmatrix} AB & -||A...

How did you find PF?: Google search How can galaxies rotate coming from a singularity? Raimund

(I know how to solve the problem, that's not what I am looking for.) I have a problem with how I ought to understand the moment of inertia. The only torque I see applicable on the wheel is that of the tension, and so I think that ##I## should be ##m_{\text{point}}R^2##, without including all the...

It's often said that you don't feel Earth rotation because the gravity acts against the centrifugal force. Of course this is true but also your body is turned around once each 24 hours. So I wonder on a planet which is rotating once each 3 seconds and has same g=9,81: Would you feel the rotation?

--no explanation as conceptual error--

I was talking to someone about the equilibrium of fluids and we reached at some stage where we had to prove that in an external field the translational forces add to zero along with moments (torques) should also add to zero. The first one was quite easy but during the discussion of second...

I have asked this question twice and each time, while the answers are OK, I am left dissatisfied. However, now I can state my question properly (due to the last few responses). Go to this page and scroll down to the matrix for sixth row of the proper Euler angles...

Let´s suppose we are observing the sun and measuring some spectral lines. Does the velocity correction´s formula for the Earth include the rotational velocity components of the Sun as well? or rather are we basically measuring both velocity contributes of Earth and of Sun together (receeding...