Rotation Definition and 1000 Threads

chapter 4.8 of Goldstein’s classical mechanics 3rd edition that deals with infinitesimal rotations, and the following is the part I got stuck: (p.166~167) : I'm not able to understand what the author is trying to say. How does "If ##d\boldsymbol{\Omega}## is to be a vector in the same sense...

Goldstein 3rd Ed pg 161. Im not able to understand this paragraph about the ambiguity in the sense of rotation axis given the rotation matrix A, and how we ameliorate it. Any help please. "The prescriptions for the direction of the rotation axis and for the rotation angle are not unambiguous...

If we change the orientation of a coordinate system as shown above, (the standard eluer angles , ##x_1y_1z_1## the initial configuration and ##x_by _b z_b## the final one), then the formula for the coordinates of a vector in the new system is given by ##x'=Ax## where...

Imagine this: You have a drum with a radius of 12cm, around that drum is a toothed belt which is connected to a motor. The drum weighs 10kg The motor should be placed under the drum How would I calculate the amount of torque needed to rotate the drum I don't have any idea how to calculate this...

The question arises the way Goldstein proves Euler theorem (3rd Ed pg 150-156 ) which says: " In three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point"...

Dear PF, so we know that cross product of two vectors can be permutated like this: ## \vec{ \alpha } \times \vec{ \beta }=-\vec{ \alpha} \times \vec{ \beta} ## But in a specific case, like ## \vec{p} \times \vec{A} = \frac{ \hbar }{ i } \vec{ \nabla } \times \vec{A} ## the cyclic permutation of...

The period of rotation of the Earth around itself is changed by seasonal winds, tides, and other factors. The exact amount changes daily(about nano second). Is there a website that shows this amount?

If I choose my axis of rotation for torque analysis to be the left-end of the plank, I can get the correct results. If I instead choose the com point -- I run into a dead end. Is there a way of a priori knowing this would happen? Thank you.

We know that ##v = \omega r## where ##r = R_{\text{E}} + h##. To compensate for the motion, the plane must fly along the equator at the same speed as the Earth but in the opposite direction, i.e. from east to west, so $$\vec{v} = -\vec{ v}_{\text{E}}$$ $$v_{\text{E}} = \omega_{\text{E}}...

If a system is represented by a set of generalized coordinates ##q_i## in which one coordinate say ##\theta## is such that ##d \theta## represents a rotation of the system about a fixed axis( an axis whose orientation remains fixed in space) then the kinetic energy ##T## shouldn't depend on it...

Magnetic fields as an alternative explanation for the rotation curves of spiral galaxies ABSTRACT THE flat rotation curves of spiral galaxies are usually regarded as the most convincing evidence for dark matter. The assumption that gravity alone is responsible for the motion of gas beyond the...

I came across an interesting question in the Hartle's textbook, "An Introduction to Eisntein's General Relativity". The question is as follows: Explain why a photograph of an object moving uniformly with a speed approaching the speed of light, parallel to the plane of the film appears not...

1) To be in equilibrium, it must be $$\begin{cases}F_{centr}-T=0\\ T-mg=0\end{cases}\Rightarrow F_{centr}=T=mg\Rightarrow m\omega^2 R_0=mg\Rightarrow R_0=\frac{g}{\omega^2}$$ 2) It is intuitive that this equilibrium is unstable but I don't know how to formally prove this. 3) In ##R_0## the...

I'm trying to model the linear collision of a bat and a ball using the conservation of angular momentum. The ball is a point particle with at rest wrt the axis of rotation, and the bat is being treated as a rod of negligible radius. I have had to work through several problems involving a ball...

I made a new version of the falling cat video, with narration. It explains how cats turn around while having zero net angular momentum during the fall:

I've a body having initial angular velocity at ## t=0 ## as shown. The axis shown are fixed in inertial space and initially match with the principal axis. I want to find the infinitesimal change at ##t+\Delta t## in the angular momentum along the ##z## axis. I've seen the following approach...

Question : Solution attempt : for

Hello, Given the figure below, and the following statement: "The robot arm is driven by two hydraulic cilinders A and B which brings point D rotates CW. The gear in point D has a angular velocity of 5 rad/s. Calculate the velocity and acceleration of the part in point C." First I determined...

I'm tutoring an intro to meteorology pupil and learning about the conservation of potential vorticity, and realizing that I don't understand some basic rotational mechanics. For example, suppose I stand on the North Pole and hold a wheel such that the wheel's axis of rotation is parallel to the...

I would like to know how to calculate, for any specified frequency in Hz, the required diameter of a 90 degree phased, quadrature coil array such that its generated EM field achieves rotation at the speed of light. Could someone please provide an example of the calculation using a specific...

Flat rotation curve in galaxies is determined by observing neutral hydrogen which is co-distributed with dark matter. What is the rotation curve profile of neutral hydrogen in galaxies where there is less dark matter?

Now, i am extremelly confused about all this thing. More preciselly, i can't understand how 1.29 was obtained. It was used the A representation? How do he uses it? There is something to do with the canonical basis?

Someone help me write a number rotation ,pleae. 1= filled ,2,3=every 14 seconds ,4=every 15 seconds,5=x 6,7=every 18 seconds,8=every 16 seconds,9=every 12 seconds,0=evrey 10 seconds, roration 300 seconds total. my bad expamle: 26783 01194 /11111 20136/ 78914 10111/ 21139 10678/41111 25031

Hi Angular momentum L is related to the moment of inertia (MOI) , I by L= Iω In the body-fixed frame , ie. rotating with the object then ω = 0 and so the angular momentum is zero in the body-fixed frame. Is that correct ? If i have a thin circular ring then the MOI about the centre is given by...

Is it correct to say that the melting of the polar caps due to climate change may increase Earth's speed of rotation?

How much will this phenomenon affect the Earth's rotation? Is it possible that the change in the Earth's rotation movement will significantly affect agricultural production on a global scale, for example?

I read this in the wiki article about Wick rotation: Note, however, that the Wick rotation cannot be viewed as a rotation on a complex vector space that is equipped with the conventional norm and metric induced by the inner product, as in this case the rotation would cancel out and have no...

Hey! :giggle: For $p\in \mathbb{R}^2$ let $\delta_{p,\alpha}=\tau_p\circ \delta_{\alpha}\circ\tau_p^{-1}$. Let $p,q\in\mathbb{R}^2$ and $\alpha,\beta\in \mathbb{R}$. (a) Show that $\gamma=\delta_{p,\alpha}\circ\delta_{q,\beta}$ is a rotation of a translation (or both). Give the center of...

Don't understand from where should I start?

At first, I inverted the function(##f^{-1}(x)=g(x)##) and calculated the volume through the integral: $$V=\pi\int_{0}^{4}[4-(2-g(x))^2]\ dx$$ but then I questioned myself if the same result could have been obtained without inverting the function. To find such a strategy, I proceeded as follows...

Can a point P on the smaller wheel trace a straight line path inside the circular road (like the walls of a well)

Inertia I = 2/5 mr² m=5.98* 10^24 kg. r=6.38* 10^6 kg. I= 9,736 * 10^37 kg. Earth rotation is V= (2 * pi* R)/T = (2*pi*6.38*10^6m)/(24*3600s)=463 m/s angular velocity w= V/r =463m/s/(6.38*10^6m)=7.27*10^-5 rad/s Enerrgy, E= 0.5* I*w²=2.57*10^29 j I get the...

The figure is shown; the measurements were taken on two consecutive observing nights. The Ordinate is the flux normalized to continuum and the abscissa is the wavelength scale. You can see the "bumps" indicated by the arrows referring to some Starspot as the spot moves on the profile; assuming a...

Someone that I tutor asked a simple but pretty good question today which I thought I'd share the answer to. In a tidied up form: a disc with centre at the origin and central axis parallel to a unit vector ##\mathbf{n}## in the ##xy## plane rotates with a constant angular velocity...

If we know the change of angle is twice the incident angle, then the rate of rotation is 2*100 rpm = 200 rpm. Is there a better explanation of it?

This was the answer key provided: My questions are the following: if the force required for rotational equilibrium is more than the limiting static friction, then the body will rotate aka slip over the surface. When it slips, the frictional force will be kinetic and not static, right? If I...

My doubt is with Method 2 of the given example in KK. I'm unable to understand why the torque around A (where we have chosen a coordinate system at A) becomes zero due to the R x F in z direction with a minus sign {Photo Attached} I have tried to reason out that one way to formulate that term...

Hi guys, Online I found this really cool experiment that uses a glucose solution(e.g. in a beaker) to rotate the plane of polarization of a polarized light beam passing through it, of an angle ##\theta## which depends on the frequency of the EM wave. Then, for example, watching white light...

Hello there I am having trouble with part b) of this exercise. I can apply the rotation matrix easily enough and get: $$ R(-\theta) \vec J= \begin{bmatrix} A\cos\theta + B\sin{\theta}e^{i\delta} \\ -A\sin\theta + B\cos{\theta}e^{i\delta} \end{bmatrix} $$ I decided to convert the exponential...

The conservation of energy equation is basically GPE is converted to KE of block and KE of cylinder. To get the correct answer, the KE of the cylinder is 1/2mv^2, where m is its mass and v is the velocity of its COM (which is the centre of cylinder). However, I viewed the cylinder as rotating...

The solution is simple by noting that the total angular momentum of the system is constant. (Though I overlooked this) Instead, I went ahead analyzing the individual angular momentum of both drums. Let ##L_a## and ##L_b## be the angular momentum respectively. ##M_a##, ##M_b## be the...

I am trying to derive the tangential acceleration of a particle. We have tangential velocity, radius and angular velocity. $$v_{tangential}= \omega r$$ then by multiplication rule, $$\dot v_{tangential} = a_{tangential} = \dot \omega r + \omega \dot r$$ and $$a_{tangential} = \ddot \theta r +...

common sense tells me that if an object can remain spinning in the vacuum of space for eternity, than placing that same object within a pressurised environment will cause the object to slow to a stop. is there a way for me to calculate the time in which it would take an object to stop rotating...

[Moderator's note: Thread spun off to allow discussion of this topic to continue since the previous thread was closed.] I have had something nagging at me about this for a while, and it finally hit me while looking through this paper about the Godel Universe...

In desmos you can rotate a label by going into the settings. BUT the rotations get weird with large numbers: if you rotate by 360 nothing happens same for 3600 and 360000 but when you get to 360000000000000000 it starts changing from just being flat. It changes from a 0 degree tilt to a 332 or...

We know that for a non-rigid body, the most stable type of rotation of it is the rotation about the axis with the maximum momentum of inertia and thus the lowest kinetic energy. However, for this question involving a rigid body, the most stable axis is the one with the lowest moment of inertia...

The rate of precession of this gyro ##\Omega## can be found by solving ##\tau_1=DMg=I_s\omega_s\Omega##. But when I apply Euler's equations to this problem, it fails. I first set the frame in the way shown in the diagram above. Then I wrote the first equation...

Hello, What I understood from multiple answers on different threads is that the effect of the time dilation is too small to explain the galaxy rotation curve. I was advised to do some calculations in order to see it myself. And this is what I would like to do but I need some help. - What is...

[Mentor Note -- Improved versions of the two pictures are posted in a reply a few posts down] Good day and here is the solution I have a problem in finding the value of AC and BC, I couldn't figure it out? many thanks in advance!