Rotation Definition and 1000 Threads

Homework Statement A small circular object with mass m and radius r has a moment of inertia given by I = cmr^2. The object rolls without slipping along the track shown in the figure. The track ends with a ramp of height R = 2.5 m that launches the object vertically. The object starts from a...

Homework Statement The probelm is to show, that a rotation matrix R, in a odd-dimensional vector space, leaves unchanged the vectors of at least an one-dimensional subspace. Homework Equations This reduces to proving that 1 is an eigenvalue of Rnxn if n is odd. I know that a rotational...

Homework Statement Block's 1 (460g) & 2 (500g) are mounted on a horizontal axle of negligible friction (R = 5.00cm). When release from rest, block 2 falls 75.0 cm in 5.00s without the cord slipping on the pulley. a) What is magnitude of acceleration of blocks? b) Tension of T2 c) Tension...

A wheel spinning clockwise on its axis at with angular velocity ω0 drops to the horizontal ground. It initially has no center-of-mass velocity. The coefficient of kinetic friction between the ground and the barrel is µ. The radius of the wheel is R, and it is a solid disc of mass M. Express your...

Homework Statement A guy is spinning on a chair with his hands at rest on his lap. As he is spinning, a large mass drops into his hands/lap. Does the guy continue spinning at the same rate, a slower rate, or a faster rate? This video demonstrates what happens when the guy drops mass...

Homework Statement A hollow shaft has an internal diameter of 72 mm and wall thickness of 24mm is to be used to transmit a power of 90 kW. what would be the maximum speed of shaft rotation if the shear stress must not exceed 150 MPa ?? Homework Equations...

Homework Statement A 0.70-kg disk with a rotational inertia given by MR^2/2 is free to rotate on a fixed horizontal axis suspended from the ceiling. A string is wrapped around the disk and a 2.0-kg mass hangs from the free end. If the string does not slip, then as the mass falls and the...

Hey, Once again, I got a question about quaternions. Say I have an initial rotation Q1. I now want to rotate Q1 on the X and then on the Y axis. BUT: The Y rotation should apply to the local Y axis which was given in Q1. The problem is: If i rotate Q1 by the X-rotation Q2, then the Y...

Homework Statement A Yo-Yo of mass M has an axle of radius b and a spool of radius R. Its moment of inertia can be taken to be MR^{2}/2 and the thickness of the string can be neglected. The Yo-Yo is released from rest. The center of the Yo-Yo descends distance h before the string is fully...

Homework Statement A disk of mass M and radius R unwinds from a tape wrapped around it. The tape passes over a frictionless pulley and mass m is suspended from the other end. Assume that the disk drops vertically. a. relate the accelerations of mass m and disk ,a and A, respectively to...

Homework Statement On the Earth the Sun appears to rise and set about 365 times in one year. During the same 365 days, how many times does the Earth rotate on its axis relative to a distant star (a star beyond the Sun and out of our solar system)? Homework Equations The Attempt...

Homework Statement A space station has the form of a hoop of radius R = 15 m, with mass M = 1000 kg. Initially its center of mass is not moving, but it is spinning with angular speed ωi = 4 rad/s. A small package of mass m = 22 kg is thrown at high velocity by a spring-loaded gun at an angle θ...

Homework Statement The problem along with its solution is attached as TheProblemAndSolution.jpg. Here is the textual part of the attached image: “In Fig. 1 a 20 ft-frame PQ is supported at two points L and M, 6 ft and 4 ft respectively from the edges. If a 300 lb load is attached to edge...

In rotation motion do points through which the axis passes also rotate or are they stationary?

Problem Statement: The system consists of a 20-lb disk A, 4lb slender rod BC, and a 1-lb smooth collar C. If the disk rolls without slipping, determine the velocity of the collar at the instant theta=30 degrees. The system is released from rest when theta= 45 degrees. Above I attached a photo...

relation between r ,ω and θ for rotation around fixed axis. \frac{d\textbf {r}}{dt} = \textbf {ω} \frac{dθ}{dt} = ω \lvert\frac{d\textbf {r}}{dt}\rvert = \frac{dθ}{dt} bold means vector. Is this right?

I am trying to visualize the following rotation of $\mathbb R^3$, but it is very difficult. I want to get the answer by intuition, and not by using the Rodrigues rotation formula or conjugation of matrices, etc. Help please. **Problem statement:** Determine the matrix that represents the...

I understand that the center of mass is a point which can be considered to contain all of an object's mass, for the purpose of calculations involving universal gravitation. I also understand that the center of mass of an object of uniform density is located at the centroid. In this case, I...

Homework Statement This isn't strictly homework, but the nature of my problem is similar to homework problems: A block with mass m and it's center of mass not necessarily at the center is sliding along a frictionless surface. The center of mass is distance r from the right front corner of...

Homework Statement Torque = 3 Time = 3.12 s Length of the each rod = 1 m so the radius = 0.500 m mass of each rod is = 0.500 kg M_1 = 4 kg M_2 = 2 kg M_3 = 4 kg M_4 = 2 kg http://imageshack.us/a/img27/5475/qu51.jpg Homework Equations What is the energy of the object (please refer...

I want to prove that: [J_1,G_1] = 0 Where J is the rotation operator and G is the boost operator (subscript refers to the axis). I am using the Jacobi identity: [[J_1,J_2],G_3] = [[G_3,J_2],J_1] +[[J_1,G_3],J_2] Using other identities, I got: [J_3,G_3] = [G_2,J_2] - [G_1,J_1]...

Hi, I would like to calculate the pressure inside a liquid planet in rotation. How can I do ? Pressure depend of depth under gravity but it depend of rotational speed too. Is it gravity pressure less centripetal pressure ? Is it \frac{1}{2}ρω^2r^2 - ρgh ?

If we consider a spin 1/2 particle, then, the rotation of the spinor for each direction is given by a rotation matrix of half the angle let say theta Rspin=\left(\begin{array}{cc} cos(\theta/2) & -sin(\theta/2)\\sin(\theta/2) & cos(\theta/2)\end{array}\right) and the new component of the spin...

Hi, I have two questions related to angular velocity: 1. According to rotational damper, Torque = Viscous Damping Coefficient * Angular Velocity. This equation gives the unit of Angular Velocity as meter square per second. How is it equivalent to rad per second? 2. If I have an angular...

Hi there, I'm new to the forum, but hopefully this question has a simply answer. my question is, suppose you have a mass (a) on the end of a string or rod attached to a vertical support which is grounded. The mass is rotating about that support. Now assume you have another mass (b) traveling...

Homework Statement A diver jumps from a 13m high tower, and hopes to complete 212 somersaults. what should be the rotation rate be Homework Equations The Attempt at a Solution 13 m = (1/2)(9.81 m/s²)(t²) t = 1.628 s ω = 2.5 rev / 1.628 s ω = 1.53 rev/s I'm still...

Homework Statement A bucket of water is swung in a vertical plane at the end of a rope of length l= 6 m. The mass of the bucket plus water is 5 kg and the gravitational acceleration is g=10 m/s2. We assume that the mass of the rope can be neglected. (a) What is the minimal speed of the...

Homework Statement The specific rotation of a tetrasaccharide was measured to be -20.5°. Upon complete hydrolysis in acid the optical rotation of the solution was found to be -36.9°. Knowing that the specific rotations of n-acetylgalacosamine,3-deoxy-l-fucose and 2-deoxy-D-ribose are...

Homework Statement A wheel of radius R=50cm rolls along the ground with velocity V=2m/s. A pebble released on top of the wheel so that it is instantaneously at rest on the wheel. The co-efficient of friction between wheel and pebble is μ=1. The pebble starts to slide down when it has rotated...

my teacher told.me.that If a rotating uniform stick is falling, the axis of rotation is about the cm of the stick. I don't understand why it is about the cm of the ball. I have searched on google and i found no answer. Can anyone give me some reference so that I can read? Can anyone explain to...

Hello MHB, As students of calculus, we are taught to find the volumes of solids of rotation obtained by revolving given regions about horizontal and vertical axes of rotation. But, what if the axis of rotation is neither horizontal nor vertical? Please consider the following diagram: We wish...

I read in several websites that triple bonds cannot rotate freely. However, I've also read in the book "Chemical Principles" the following lines: "Various types of evidence suggest that the electron density around the two C-O bonds in CO2 is actually cylindrically symmetric—that is, the...

How exactly can the angular velocity of a 2 dimensional laminar object be the same with respect to all axes of rotation perpendicular to its plane ?

Is it possible to check the phase rotation sequence of a Medium Voltage Motor (Up to 6 kV) with one instrument? Does the shaft need to be moving while it is being tested? If instead of one, there are two motors electrically connected in parallel, with a common source feeder. will it be...

Hello everyone! This is probably the stupidest question that I've come up with, and I'm a little embarrassed asking it, but here goes: Is it only the tangential speed that is zero of a point at the axis of rotation in a rotating solid? If not, then I don't understand how the rotational...

If one constructed a platform at the absolute south pole, that is truly perpendicular to the Earth's axis and then constructed a shaft on such platform that is aligned parallel and on center with the Earth's south pole axis and then installed a bearing, with large weights on the outer...

Hi, Is there any good books which explain/calculate Wigner rotation, tensor operator, two-particle helicity state and related stuff in detail? Thanks.

Homework Statement Which matrix represents a rotation? Homework Equations The Attempt at a Solution It seems odd that this matrix has somewhat the form for rotation about z-axis, just that you need to swap the cos θ for the sin θ.

Hi friends, Feeling pleasure to share some more aspects with you. I am getting a doubts in some of the problems. Thank you all in advance. The problem is: Attempt, But the book says the answer is option (D). Please friend help to get rid off. Awaiting for reply.

Hi guys, This isn't really a homework problem but I just need a bit of help grasping rotations in spherical coordinates. My main question is, Is it possible to rotate a vector r about the y-axis by an angle β if r is expressed in spherical coordinates and you don't want to convert r...

As you may know, Foucault's pendulum is an easy way to verify that the Earth is rotating about its axis. Its a pendulum that is free to swing in any direction. Since Earth rotates under it, the position of the pendulum with respect to the ground changes after some time. So you could put pins on...

Hey, Got into a discussion with my friends over this: If you spin in your chair clockwise, since the Earth spins counterclockwise, as you start spinning by conservation of momentum the Earth would also start speeding up. Then, frictional dissipative forces from the air would slow you down to a...

If I have a 45 degree linear polarized light which I then circularly polarize using a 1/4 wave plate and put this through an optical rotary crystal and then using the equivalent 1/4 wave plate but in the reverse oriention, will I get back a 45 degree linear polarized light? Put another way...

Homework Statement Mass m is attached to a cylindrical post of rdius R by a string. Initially it is distance r from the centre of the post and is moving tangentially with speed v_{0}. In case (a) the string passes through a hole in the centre of the post at the top. the string is...

Homework Statement I want to derive the trig identities starting with rotation on the plane. Homework Equations One rotation through a given angle is given by $$x' = xcosθ - ysinθ $$ $$y' = xsinθ + ycosθ$$ The Attempt at a Solution What if I wanted to rotated through any...

I have a question about Earth's rotation around its own axis. Earth rotates at around 1000 miles per hour. However, if I stand in an open field and let a balloon float beside me in mid-air, I wouldn't expect the balloon to fall away (or appear to fly away) at that speed when I let go of it. I...

Homework Statement a drum of mass M_{a} and radius a rotates freely with initial angular velocity ω_{a}(0). A second drum with mass M_{b} and radius b greater than b is mounted on the same axis and is at rest, although it is free to rotate. a thin layer of sand with mass M_{s} is distributed...

If that happened, Will it affect to the gravity?

Homework Statement A rotation ρ about the origin in ℝ2 drives the point P = (4,3) to the point ρ(P) = (3,4). Find the angle of rotation as well as its matrix notation. Homework Equations Ok so I made a sketch and I realized I needed to find θ = θ1 - θ2 where θ1 and θ2 equal arctan(4/3) and...

How would a galaxy rotation curve look if every matter simply had a 6 times larger mass than the visible? (please neglect how that could be) Wouldn't a same size galaxy then reside in a 6 times larger gravitational well so that the spiral arms would still be in the steep part of the well...