Rotation Definition and 1000 Threads

A surface is obtained by rotating around the x-axis the arc over the integral(-1,0.5) of an ellipse given by: x^2+4y^2=1 What is its surface area? Here's my solution: I use the equation: S=integral( upper bound: a lower bound: b ) 2(pi)y*[1+(f'(x))^2]^0.5 dx Since x^2+4y^2=1...

There is a rigid solid sphere rolling without slipping on a horizontal surface. When we are taking the CM as the axis of rotation, we see that the Total Kinetic Energy comes out be as: KEtotal = KErotational + KEtranslational Here: KErotational = (1/2)ICMω2 where ICM is the Moment of Inertia of...

Homework Statement A disk rotates with angular velocity ω. It has a groove cut along the diameter in which two blocks of mass m and M slide without friction. They are connected by a light string of length l, fixed by a catch with block m a distance r from the center (r + radius of M = l)...

Homework Statement Prove that a rotation matrix in R3 preserves distance. Such that if A is a 3*3 orthogonal rotation matrix then |A.v|=|v|. I know one can prove this is in R2 by using a trig representation of a rotation matrix and then simplifying. Is there an analogue method in R3 or some...

for a perfectly rigid body, how can one identify what is the axis of rotation of the rigid body? What is the condition required for an axis to be called the axis of rotation?

Hello. My first post here. I'm having trouble with the basics of rigid body rotation. I have a few questions (my apologies if they are too childish; I'm very new to this): 1) Is torque (and other angular parameters like angular velocity, angular acceleration etc.) defined about a point or an...

Homework Statement Let C be the circle defined by (x-2)2+y2 = 1. If this circle is rotated along the y-axis, a torus will form. What is the Cartesian equation for the torus? The Attempt at a Solution The solution manual says you just switch the x in (x-2)2+y2 = 1 with r=√(x2+z2) and...

From "Modern Quantum Mechanics, revised edition" by J.J. Sakurai, page 196. Equation (3.6.4), 1-i \left( \frac{\delta \phi}{\hbar} \right) L_z = 1 - i \left( \frac{\delta \phi}{\hbar} \right) (x p_y - y p_x ) Making this act on an arbitrary position eigenket \mid x', y', z' \rangle...

I was asked to formulate the equations governing the rotation of a body moving without any external moments acting about its centre of mass in terms of a coupled system of first order, nonlinear differential equations. I decided to go with the Euler equations, and I ended up with this...

Hello MHB, Do anyone know any good page that give you good describe when you rotate with orthogonal. I mean when you rotate base or vector in a orthogonal base ( hope this make sense) cause I did not understand from my book :( Regards, $$|\pi\rangle$$

I'm wondering if the following is possible. Consider some inertial coordiante system x, y, z, and a rotating coordiante system p, q, r defined through matrix rotations as follows. \begin{pmatrix} p \\ q \\ r \end{pmatrix} = R_1(\theta_1(t)) R_2(\theta_2(t)) R_3(\theta_3(t))...

I'm working on an inverse kinematics problem (I make video games), and I'm reaching a bit beyond my education. Right now, I've got an algorithm that solves the basic IK equation for a chain of rigid bodies connected by joints by approximately inverting ##J\Delta\theta = e##. Where ##\theta##...

A chimney is demolished by breaking its base in hope that when it falls the top of the chimney will prescribe a circle. If the chimney is broken into two segments in the air, will the top segment reach the ground first? I have attached my professor's solution. My professor didnt...

1. Problem statement A simple model of a crane involves a motor driven drum that winds or unwinds a steel cable onto the drum. The cable passes through a pair of guides and over a pulley at the end of the crane arm. Ignore the mass of the cable in your calculations. The drum has a mass of...

I'm trying to rotate a point about the origin (0,0,0) and starting with an identity matrix, this works fine for the x- and y-rotation axes, but fails with the z-axis, where the point just sits in place. \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} M_{ID} \times M_Z...

Hi everyone, I'm stuck on the concept of the rotation operator in QM. From what I understand, one constructs a representation of SO(3) on a Hilbert space by mapping a rotation matrix R\in SO(3) specified by an angle \phi and a unit vector \vec{n} to D(R) = \exp[-\frac{i...

define curl "rotation per area" When they define curl, they say it is a measure of "infinitesimal rotation", or "rotation per area". What does that mean? Does it mean they measure how much something goes around in an infinitesimal point (which makes no sense), kind of like a whirlwind shrunk...

So I am given B=\begin{array}{cc} 3 & 5 \\ 5 & 3 \end{array}. I find the eigenvalues and eigenvectors: 8, -2, and (1, 1), (1, -1), respectively. I am then told to form the matrix of normalised eigenvectors, S, and I do, then to find S^{-1}BS, which, with S = \frac{1}{\sqrt{2}}\begin{array}{cc} 1...

Calculate the area of ​​the surface of rotation which occurs when the curve rotate in y-axe. I start with $$x=\sqrt{28y}$$ then $$f'(x)=\frac{14}{\sqrt{28y}}$$ so we got $$2\pi\int_0^{21}\sqrt{28y}\sqrt{1+(\frac{14}{\sqrt{28y}})}^2$$ then I rewrite as...

A thin, cylindrical rod = 26.6 cm long with a mass m = 1.20 kg has a ball of diameter d = 10.00 cm and mass M = 2.00 kg attached to one end. The arrangement is originally vertical and stationary, with the ball at the top as shown in the figure below. The combination is free to pivot about the...

When solving for instanton solutions in a 1+1d abelian Higgs model, it's convenient to work in Euclidean space using the substitution x^0 \rightarrow -ix_4^E,\quad x^1 \rightarrow x_1^E The corresponding substitution for the covariant derivative is D^0 \rightarrow iD_4^E,\quad D^1 \rightarrow...

Earth is completing one rotation in one day i.e, in 24 hours. Its equator length is around 40,000 km. Every point on equator is moving with a speed of 40,000/24 i.e, 1667 km/hr. Why don't we feel that speed even though we are moving at such high speeds when we are at equator?

sorry, this is is a general question about a conceptual definition I read in my textbook, i hope that's ok. "an object that rolls without slipping at a constant velocity over a surface with friction experiences no frictional force" is this true? i understand that on a frictionless surface, the...

I have some doubts on topics related to rotation, so I thought I'd make a single thread rather than multiple ones. 1. Why are the Coriolis and Centrifugal forces "fictitious"? I think I might be getting confused by the terminology, but do they represent physical forces? My lecture notes say...

The energy available at room temperature is 0.593 kcal/mol (wikipedia) so why is it that Ethane is said to freely rotate from staggered to eclipse if it has a rotational energy barrier of 2.9 kcal/mol (wikipedia)? What am I missing here?

Homework Statement A coin is spinning on its edge at 5 rotations per second. Friction slows down its spin rate at .4 r/s2 a) what angular displacement does the coin have by the time it's slowed down to half its original angular velocity. b) how long before the coin stops spinning?

Homework Statement A merry - go - round rotates clockwise 15 times in 45 seconds. By rubbing against its edge, a child stop it from turning in 15 seconds. a) find its initial angular velocity (ω) b) find its angular acceleration (\alpha) The Attempt at a Solution 15 times in 45...

Homework Statement A uniform circular disc has mass M and diameter AB of length 4a. The disc rotates in a vertical plane about a fixed smooth axis perpendicular to the disc through the point D of AB where AD=a. The disc is released from rest with AB horizontal. (See attached diagram) (a)...

(If this is in the wrong section, please feel free to move it.) Hi all, How would one calculate the equatorial rotation velocity or rotation period without the other? Is this possible using the values; diameter, circumference, mass and revolutionary/orbital period? Thank you.

Hello, I see that hyperbolic rotation of frame F' about the (x2,x3)-plane of frame F is identical to a Lorentz transformation, corresponding to a linear motion along x1 of the frame F' with respect to F. Then hyperbolic rotation about (x1,x2) means motion along x3 and hyperbolic...

Homework Statement A ball moving in the circular path with a constant speed of 3.0 m/s changes direction by 40.0 degrees in 1.75 seconds. What is the change in velocity? What is the acceleration during the time? Homework Equations Fc = m * ac ac = v^2/r The Attempt at a Solution...

Hi, 1) Do we calculate torque with respect to a point or with respect to an axis? I have read them both in different resources, and so I am confused! 2) If we calculate torque with respect to an axis, many introductory textbooks discuss the motion of the gyroscope by considering how the...

Homework Statement A turntable of radius "r" is spinning counterclockwise at an initial rate of ω. at t=0, its rotation rate begins to slow at a steady pace. the rotation finally stops at t=T. At what time is ωt^2 maximum. Express your answer in terms of T.

Hello, Can somebody please tell me in details about the rotation speed of a neutron star? Does it rotate very fast and then slows down? Thanks.

I have a uniform grid of data in spherical coordinates. e.g. theta = 0, 1, 2, ... 180 and phi = 0, 1, 2, ... 359 which forms a 2D matrix. I wish to rotate these points around a cartesian axis (x, y, z-axis) by some angle alpha. To accomplish this I currently do the following: 1. Convert to...

Homework Statement consider the following rotation matrix: 0 0 1 1 0 0 0 1 0 Find the axis of rotation. Homework Equations The Attempt at a Solution I know the following: Ω|1> = |2> Ω|2> = |3> Ω|3> = |1> where Ω is an operator. It is a cyclic permutation. What do not understand is...

Hi! I was watching Discovery channel the other day and it was showing a series on evolution of guns and ammo.. It showed that earlier there was bolt mechanism for firing the gun. And then, a revolution came when muzzzle or bullet was shot such that it went in forward direction while...

The telescopic boom of a crane rotates with the angular velocity and rotation as indicated about point $A$. At the same instant, the boom is extending with a constant speed of 0.5ft/s, measured relative to the boom. Determine the magnitude and acceleration of the absolute acceleration of point...

I am utilitizing rotation vectors (or SORA rotations if you care to call them that) as a means of splitting 3D rotations into three scalar orthogonal variables which are impervious to gimbal lock. (see SO(3)) These variables are exposed to a least-squares optimization algorithm which...

Hello all, I'm currently a undergrad university student doing research and I'm anaylising some position data. The data is a time-series' of Eastings (x) and Northings (y) for two points (P1, P2) on a rigid body in motion (T, E1, N1, E2, N2), with a position reported every 1 minute. I know...

For a mathematician, how is rotation defined in the most general sense? Question arose to me because it occurred to me that an essential property of the rotation matrix is that it preserves lengths. Is this the only mapping (if not please give me a counterexample) that has this property...

Since the redistribution of mass in the Earth's surface can be caused by earthquakes, sometimes the Earth's rotation is increased or decreased by a small amount. Recent series of quakes seem to be related. To me, this makes sense, since if plate "A" should move, then plate "B" would also...

hi, I understand how to do the rotation equation A = [ cosθ -sinθ sinθ cosθ] A*v = [ cosθ -sinθ * [ x = [ xcosθ - ysinθ sinθ cosθ] y ] xsinθ + ycosθ ] A*v = [ cos90 -sin90 * [ 6 sin90 cos90 ] 4 ] =...

How does finding the rotation operator for a spin 1/2 particle differ from finding that of a spin 1 particle?

In the framework of Classical Mechanics,there is no problem in the rotation of rigid body.But I want to make the concept about rotation more clear. About rigid body ,there is no vibration;it would only rotate and translate.We can easily distinguish translation from rotation in rigid...

Homework Statement On a X-Y plane we have a square with its 4 corners A(3,1) B(7,3) C(2,6) D(0,2). We are to rotate the rest of the square around the point A clockwise by 70 degrees. Homework Equations (I am not sure how they are called in English) The rotation matrix 2x2 1st row...

Hello, Do stars rotate? I mean to say that do they rotate in angular momentum? Or there is any other rotation?

Hi, This is one of the things that confuses me. Assume an object on the surface of the Earth has a mass m and F=m*g is the force on it due to the gravity. But also the Earth is rotating and although the radius is extremely large compared to the size of the object but it must be affected...

Homework Statement Homework Equations Shell method The Attempt at a Solution Not sure if this is right, but the integral I set up is: 2\pi \int_0^1 (4-y)(\sqrt{siny})dy Finding the radius is my big problem with these problems, I can't visualize it very well. Any help is...

For a 2D vector field {F}=P(x,y)\vec{i}+Q(x,y)\vec{j} curl {F} = \frac{\partial Q}{\partial x}+\frac{\partial P}{\partial y}\vec{k} So that's the rate of change of the j component of a field vector with respect to x plus the rate of change of the i component with respect to y...how does...