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[NOTES/CM-09005] Heavy Symmetrical Top with One Point FixedNode id: 6236page$\newcommand{\Label}[1]{\label{#1}}\newcommand{\eqRef}[1]{\eqref{#1}}\newcommand{\Prime}{{^\prime}}\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}} \newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}}\newcommand{\dd}[2][]{\frac{d#1}{d #2}}$ |
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24-05-14 08:05:05 |
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[NOTES/CM-09004] Kinetic Energy of Rigid BodyNode id: 6235pageAn expression for the kinetic energy in terms of the moment of inertia tensor and the angular velocity w.r.t the body frame of reference is obtained. It is shown that \begin{equation} \text{KE}=\sum_{ij} \omega_{bi} I_{ij}^{(b)} \omega_{ij} \end{equation} |
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24-05-14 05:05:01 |
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[DOC/LOG-2024-03] LOG, HISTORY and INFORMATIONNode id: 6086pageThe workflow, time allotment to different activities of Proofs program are divided into several compartments.
This is a log file |
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24-05-14 04:05:22 |
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[NOTES/CM-09001] Degrees of Freedom of a Rigid BodyNode id: 6234page |
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24-05-13 16:05:52 |
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some title hereNode id: 6233collection |
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24-05-13 05:05:10 |
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[NOTES/CM-08001] The Group of Special Orthogonal Matrices in \(N\) DimensionsNode id: 6232page$\newcommand{\Label}[1]{\label{#1}}\newcommand{\eqRef}[1]{\eqref{#1}}\newcommand{\U}[1]{\underline{\sf #1}}$
The definition of special orthogonal group in \(N\) dimensions is given. For \(N=3\) this is just the group of all proper rotations. |
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24-05-10 05:05:05 |
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[DOC/Latex/Macros] Freq-Used-MacrosNode id: 6231page |
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24-05-10 05:05:03 |
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[NOTES/CM-08013] Rotation of a Vector about Coordinate AxesNode id: 6230pageWe derive the transformation rules for a rotation about \(X_3\)- axis. The concept of active and passive rotations is briefly explained.$\newcommand{\Label}[1]{\label{#1}}, \newcommand{\Prime}{^\prime}\newcommand{\eqRef}[1]{\eqref{#1}}$ |
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24-05-09 22:05:49 |
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[NOTES/CM-08014] Active and Passive RotationsNode id: 6229pageThe active and passive view of rotations are defined. |
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24-05-09 12:05:46 |
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[NOTES/CM-08010] Motion in Frames with Linear AccelerationNode id: 6223page$\newcommand{\Prime}{{^\prime}}\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}} \newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}} \newcommand{\dd}[2][]{\frac{d#1}{d #2}}\newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}}$
The equations of motion in a linearly accelerated are are derived and an expression for pseudo force is obtained. |
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24-05-09 12:05:47 |
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[NOTES/CM-08003] Euler AnglesNode id: 6216page$\newcommand{\Prime}{^{\prime}}$
Euler angles are an important ways of parametrization of rotations. The definition of Euler angles and and expression of the rotation matrix in terms of the Euler angles are given. |
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24-05-09 12:05:39 |
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[NOTES/CM-08006] Axis Angle Parametrization of Rotation Matrix Node id: 6218page$\newcommand{\Prime}{{^\prime}}\newcomman{\U}[1]{\underline{\sf #1}}$
A closed form expression for rotation matrix is derived for rotations about an axis by a specified angle \(\theta\).
\begin{equation} R_{\hat{n}}(\theta)=\widehat {Id}+\sin (\mu \theta) (\hat{n}\cdot \vec{I})+(1-\cos \mu \theta)(\hat{n}\cdot\vec{I})^{2} \end{equation}Here \(\widehat{Id} \) is the identity matrix. and \(\vec{I}=(I_1,I_2,I_3)\) is given by \begin{equation} I_1=\left[\begin{array}{clc} 0 &0 &0\\ 0 &0 &-1\\ 0 &1 &0 \end{array}\right],I_2=\left[\begin{array}{clc} 0 &0 &1\\ 0 &0 &0\\ -1 &0 &0 \end{array}\right],I_3=\left[\begin{array}{clc} 0 &-1 &0\\ 1 &0 &-1\\ 0 &1 &0 \end{array}\right] \end{equation}.
Also the components of the position vector a point transform a
\begin{equation} {\vec{x}}\Prime=(\hat{n}\cdot{\vec{x}})\hat{n}+\cos\theta\big(\vec{x}-(\vec{x}\cdot\vec{n})\hat{n}\big)-\sin\theta (\hat{n}\times \vec{x})\end{equation} |
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24-05-09 11:05:36 |
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[NOTES/CM-08008] Proper Rotations and $SO(3)$Node id: 6219page$\newcommand{\Prime}{{^\prime}}\newcommand{\Label}[1]{\label{#1}}$
The definition and properties of proper rotations are presented. |
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24-05-09 10:05:41 |
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[NOTES/CM-08004] Motion in Non Inertial FramesNode id: 6220page
$\newcommand{\U}[1]{\underline{\sf #1}}\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$
We derive an expression for Lagrangian for motion of a charged particle in a rotating frame, It is shown that the equation of motion can be written as
\begin{eqnarray} m\ddot{\vec{x}}=\vec{F_{e}}-2m\vec{\omega}\times{\dot{\vec{x}}}-m\vec{\omega} \times(\vec{\omega}\times\vec{x}) \end{eqnarray} where \(\vec {F}_e\) is the external force. As seen from the rotating frame, the particle moves as if it is under additional forces
- $-2m\vec{\omega}\times{\dot{\vec{x}}}$ is called Coriolis force
- $-m\vec{\omega}\times(\vec{\omega}\times\vec{x})$ is known as centrifugal force
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24-05-09 06:05:58 |
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[NOTES/CM-08009] Effect of Earth's RotationNode id: 6222pageThe effect of Coriolis force on the force acting on a body on the earth is computed at the poles, equator and at a general point. he banking of railway tracks and of roads is briefly discussed. |
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24-05-09 04:05:54 |
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[WHER/CM-07001] Where does Harmonic Oscillator Appear In Engineering ?Node id: 6228collection |
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24-05-06 04:05:08 |
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[WHER/GT-08001] Where do Rotations Appear in Physics?Node id: 6226page |
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24-05-05 08:05:05 |
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[WHER/GT-ALL] Group Theory >> Where does a Topic Appear in Physics and ...?Node id: 6227collection |
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24-05-05 07:05:55 |
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[DOC/RESOURCES-ALL] Classification of ResourcesNode id: 6099page |
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24-05-05 07:05:16 |
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[NOTES/CM-08012] Matrices for Rotations about Coordinate AxesNode id: 6225pageThe rotation matrices for rotations about the three axes are listed. |
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24-05-03 08:05:13 |
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