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.
Hello everyone!
I was wondering why can't we take a rotating body and see the linear movement that each particle moves to find the 'total linear momentum,' I imagine this quantity would be conserved, and furthermore couldn't you write the total linear momentum as a function of angular velocity...
If you put a hockey puck on a flat Ice rink, will it move due to the Earths rotation? For example, if I make a mark and measure the distance moved with a caliper, would I notice a change?
I have been trying to determine an expression for a unit vector in the direction of F for hours now.
I think the expression is supposed to look something kind of like this,
But I don't understand at all how to arrive at this expression.
Any help?
Hey, I have a question about proving Saha's equation for ionizing hydrogen atoms.
The formula is
\frac{P_{p}}{P_{H}} = \frac{k_{B} T}{P_{e}} \left(\frac{2\pi m_{e} k_{B}T}{h^2} \right)^{\frac{3}{2}}e^{\frac{-I}{k_{B} T}}
with
P_{p} pressure proton's,
P_{H} pressure hydrogen atoms,
m_{e}...
I'm not too sure how to account for both the mass and the rope at once.
I think the following are true for the two individually:
For the mass at the end, ## T = m ω^2 L ##, following from ##a = v^2/r##and ##v=ωr##.
For the rope, ##dT = ω^2 r dM##, where ##dM = λ dr## and λ is the mass per unit...
Ok. So, I already worked on this problem, and get ##m_c## = 2m/3, which is correct according to the book.
However, I also want to know the value of the tension (T) between rod A and B.
Note: Before we start working on my modified question, I want to point out that the force exerted by the...
Suppose the car is moving to the right, so if the wheels roll without slipping, they are rolling clockwise. To get the wheel to slip, a counterclockwise torque would need to be applied to cause the wheel to have some angular acceleration. If the wheel was slipping, then the bottom of the wheel...
I am doing a University lab project where I measure positions of sunspots (using images from NASA's SDO) and use them to calculate the rotation of the Sun. Currently, all is going well: I have the angular velocity of several sunspots at varying heights. However, I want to be able to find the...
Hello, so I have a question about the sense of rotation of the body.
I get the calculating part nd stuff like that. But what I don't understand is how we would determine the sense of rotation about the moment axis?
Could someone explain this to me please? (to add to this, I know that it is...
When a cyclist rides off a ‘drop’ (an abrupt step in topography, ranging from a curb to a cliff), the front wheel starts falling before the back wheel, so that by the time the back wheel comes off the drop, the bike will not be horizontal. The front wheel will be lower than the back wheel by...
figure 11.12
I need someone to explain why the angular momentum of the ball is ## L_{f} = -rm_{d}V_{df} + I\omega## rather than ## L_{f} = rm_{d}V_{df} + I\omega ##. How to distinguish the sign of the angular momentum?p.s. ##\Delta\vec{L}_{total} = \vec{L}_{f} - \vec{L}_{i} = (-rm_{d}v_{df} +...
I'm now learning about rotational motion without slipping and it's really hurting my brain to think about. Imagine a cylinder rotating on a flat plane.
I can accept that there is both translational and rotational motion. For example, a given point on the circumference of the cylinder follows a...
While revising Rotational motion, I came across a qualitative question which blew me away. Meaning I couldn't even understand the question let alone answer it😅. It has to do with these objects called spinors which as I understand are evoked in quantum mechanics and Relativity. I am attaching the...
I'm currently working on a pet project which is similar to the OpenAI Lunar Lander v2 (hence the problem is in a 2D context), and seeking help for a sub-problem that's been blocking me for a while.
At any instant of time, I'm to find
Fe: magnitude of main engine thrust, must be >0
Fs...
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...
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?
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...
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...
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...
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...