Rigid body rotation Definition and 33 Threads

Problem: Official solution: My calculation: \begin{align*} \mathbf L &= M \mathbf R \times \mathbf V + \mathbf L_{cm} \\ &= M R (\hat j + \hat k) \times (- \Omega R \hat i) + MR^2 \Omega \hat j \\ &= MR^2 \Omega (\hat k - \hat j + \hat j) \\ &= MR^2 \Omega \hat k \end{align*}

Consider the above setup. Here, to get the kinetic energy of the body, the moment of inertia with respect to the ##y-##axis has to be calculated. This can be done in two ways: 1. The moment of inertia of the rotation around the center of mass is ##\Theta_s##, then the kinetic energy is...

For a rigid body with three principal axis with distinct moments of inertia, would the principal axis with the intermediate moment of inertia still be unstable in ideal conditions, e.g. no gravity, no friction etc.? From the mathematical derivation I deduce that it should be unstable, since we...

So I have 2 Points P1 and P2. P2 is the center of mass which has an X and Y coordinate. P1 is where I think that it will fall over from and only has an X location. Now what I want is to find a way to get how far the angle at P1 will change in relation to where these points are within a given...

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...

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

(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...

I have been studying the dynamics of free top from Morin's book. In his book when describing the dynamics,he writes down the equation of motion as shown in screenshot. However,I am not able to understand which term refers to which coordinate system. For eg: Here ##\omega## refers to angular...

This is a question about the concepts behind rigid body rotation when we use relative velocity. In general, let us say that we have a rigid body and on it are two points, A and B, which are moving with velocities vA and vB respectively. These velocities are in random directions. The theory...

I am trying to solve problem number 4 part A and B from (http://people.physics.tamu.edu/kamon/teaching/phys218/exam/2003C/2003C_Exam3_Solution.pdf) but I am confused about certain aspects of it. In part A, I understand that since we are considering the person as a cylinder, the equation for...

Just to confirm a few points: A body frame rotates with a body. It need not be aligned to the rotation axis. Angular momentum vector always aligned with rotation axis (not deviates from it). An observer on the body surface (body frame) observes no motion on a body. Principal axis of rotation...

I have 2 ellipsoids: Ax^2/a^2+Ay^2/b^2+Az^2/c^2=1; (*a,b,c>0 constants*) Bx^2/a^2+By^2/b^2+Bz^2/c^2=1; Ellipsoid A rotates around axis [wax;way;waz] (unit vector) with an infinite speed; Ellipsoid B rotates around axis [wbx;wby;wbz] (unit vector) with an infinite speed; [Dx;Dy;Dz] is the vector...

Suppose we have a rigid body that rotates around a point P other than its center of mass, with point P being a point of the rigid body. This implies that there are external forces to the rigid body. If the external forces cease to exist(like we have an axis passing through P and we suddenly...

a disc of mass m and radius r rolls down a slope of incline tan^-1 3/4. the slope is rough enough to prevent slipping. the disc travels from rest a distance of s metres straight down the slope. (i) show that the linear acceleration of the disc is 2/5 g. what i did was i first revolved the forces...

The main question is in the title. 1.I have wondered if a regular pencil would precess and rotate stable for multiple seconds if spun fast enough. [Coins, plates, bowls and similarly shaped objects can rotate in a stable way for extended periods of time. And a lot of other household objects...

For the rotation of a rigid body about a fixed axis z the following holds. $$\vec{\tau_z}=\frac{d\vec{L_z}}{dt}= I_z \vec{\alpha} \tag{1}$$ Where \vec{\tau_z} is the component parallel to the axis z of a torque \vec{\tau} exerted in the body; \vec{L_z} is the component parallel to the rotation...

Homework Statement Consider the rigid body in the picture, rotating about a fixed axis z not passing through a principal axis of inertia, with an angular velocity \Omega that can vary in magnitude but not in direction. Find the angular momentum vector and its component parallel to z axis (...

This is of my own interest/ practice. Homework Statement A thin rod (of width zero, but not uniform) is pivoted freely at one end about the horizontal z axis , being free to swing in the xy plane (x horizontal, y vertically down). Its mass is m and its CM is a distance a from the pivot. The...

Hi, I am having some problems conceptualizing the Euler's Theorem. Any help will be greatly appreciated. In Goldstein's book the Euler's theorem is stated as 'Any displacement of a rigid body, whose one point remains fixed throughout, is a rotation about some axis', then he has proven that the...

in the figure a rigid body - a circle- is moving such that its centre is moving in a circular path but the orientation of the body is fixed with respect to the centre of the body (the circle). According to def of rotion of rigid body - Rotation of a rigid body about a fixed axis is defined as...

Hi everyone, I was recently talking to someone with a non-maths background about rotational stability, in particular how rotation is stable around the largest and smallest principal moments but not the intermediate one. He asked me if there was any 'obvious' reason for this, but one didn't...

1. The problem statement, all variables and given/know Say I have a can of water, and I am rotating it about its central axis at a constant angular rate. The water in the tank should make a 3D almost parabolic curve as it touches the the walls of the tank. Can I use Bernoulli's equation along...

Hi, I have a question about rigid body rotation. If I have a rod out in space in free float and I attach a battery powered fan to center of mass and then the fan is turned on to spin in a clockwise direction, then the rod should spin in a counterclockwise direction. This much I...

Homework Statement This is not a HW problem, but something I am trying to prove/disprove for my own knowledge. I have a tube pinned at both ends and inclined in the x-y plane. The pin locations can both move freely in space but subject to the constraint that the tube will not stretch or...

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 Equate the expression for centripetal acceleration with the gravitational acceleration to show that the central parts of the galactic rotation curve are consistent with rigid body rotation. The attempt at a solution Say a star near the galactic center has mass m and the...

Homework Statement https://dl.dropbox.com/u/66988308/Capture.JPG Homework Equations V=rWThe Attempt at a Solution I tried to plug in the forumla , and algebras but I keep getting rc/rf as answer instead of rf/rc as the book says. Work: Vf=(rf)W W=Vf/rf W=Vc/rc Vc/rc = Vf/rf (Vc)(rf) = (Vf)...

Homework Statement A system is made of the following things: A homogeneous disk of mass m = 4kg. Having radius r = 80cm's. A point P (m = 2kg) free to rotate around a horizontal axis which is perpendicular to the disk in point A. AO = OP/2 If the system starts from rest and AP forms a...

Homework Statement I'm trying to include rigid body rotation in a problem I'm working on but can't seem to figure it out. Two shafts oriented vertically are connected by a thin cross member of length R. Holding one shaft stationary and applying a constant tangential load F to the other...

Note this is physics I This should be the right section as this is not homework.. Ok I'm having trouble understanding the concepts of rolling without friction, kinetic energy linear and kinetic energy rotational. I have a hard time following my professor in class and usually like to go...

Homework Statement imagine a thin length of wood supported by two supports, one at either end. One support dissapears and the other turns into a pivot (as gravity acts on the wood). Show that the load supported by the pivot is Mg/4. The Attempt at a Solution i don't know where to...

a plank is conected horizontally at both ends to two points. One of the points dissapears and the other turns into a pivot. show that the Planck of length l,mass m pivoted at one end has an acceleration (3/2)*g initially i used parallel axis theorm do derive the moment of inertia of a...

I have found many cases of fluids entering rigid body motion where the gravity vector is purely down the rotation vector. I am curious if there is a soultion for where the gravity vector is in another direction. I've attempted to solve this myself for a particular rotation, but it is so fast...