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[NOTES/CM-09014] Angular Momentum of a Rigid BodyNode id: 6244pageThe angular momentum of a rigid body is given by where \(\mathbf I\) is moment of inertia tensor and \(\vec \omega \) is the angular velocity.\begin{eqnarray} \vec{L}&=&\int dv \rho(\vec{X})\vec{X}\times(\vec{\omega}\times\vec{X})\\ &=&\int dV \rho(\vec{X})\Big[(\vec{X}\cdot\vec{X})\vec{\omega}-(\vec{X}\cdot\vec{\omega} )\vec{X}\Big] \end{eqnarray} or \(\vec L=\mathbf I\, \vec \omega\). |
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[NOTES/CM-09008] Specifying Orientation Using Body AxesNode id: 6238pageA possible way of specifying the orientation of a rigid body is to give orientation of body fixed axes w.r.t. a space fixed axes. Euler angles are a useful set generalized coordinates to specify orientation of the body axes relative to a space fixed axis. |
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[NOTES/CM-09001] Degrees of Freedom of a Rigid BodyNode id: 6234pageBy considering possible motions of a rigid body with one, two or three points fixed, we show that a rigid body has six degrees of freedom. |
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24-05-15 07:05:56 |
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[NOTES/CM-09005] Heavy Symmetrical Top with One Point FixedNode id: 6236page$\newcommand{\Label}[1]{\label{#1}}\newcommand{\eqRef}[1]{\eqref{#1}}\newcommand{\Prime}{{^\prime}}\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}} \newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}}\newcommand{\dd}[2][]{\frac{d#1}{d #2}}$
We set up Lagrangian for a heavy symmetrical top and show that the solution can be reduced to quadratures. |
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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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[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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[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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[WHER/CM-07001] Where does Harmonic Oscillator Appear In Engineering ?Node id: 6228collection |
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[WHER/GT-08001] Where do Rotations Appear in Physics?Node id: 6226page |
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[DOC/RESOURCES-ALL] Classification of ResourcesNode id: 6099page |
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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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[NOTES/CM-08002] An explicit form of rotation matrixNode id: 6214pageFor two sets of coordinate axe \(K\) and \(K^\prime\) having common origin, an explicit form of the rotation matrix connecting them is obtained in terms of direction cosines. |
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24-05-03 04:05:37 |
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[NOTES/CM-08005] Finite Rotations of Vectors about an Arbitrary AxisNode id: 6217page$\newcommand{\Prime}{{^\prime}}$
Using geometrical arguments, we will derive the result \begin{equation}\vec{A}^\prime = \vec{A} - (\hat{n}\times\vec{A})\, \sin\alpha + \hat{n}\times (\hat{n}\times\vec{A})\, (1-\cos\alpha ) \end{equation}between components of vectors related by a rotation by and angle \(\theta\) about an axis \(\hat n\). |
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24-05-02 20:05:20 |
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[NOTES/CM/08001] The Group of Orthogonal Matrices in Three DimensionsNode id: 6215pageThe groups of all orthogonal matrices is defined It has a subgroup of matrices with determinant +1, This subgroup is called specail orthogonal group. $\newcommand{\U}[1]{\underline{#1}}$ |
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24-05-01 06:05:13 |
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[NOTES/CM/The Group of Special orthogonal Matrices Three Dimensions]Node id: 6212page |
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[LSN/EM-ALL] Electromagnetic Theory ---Stockpile of Lessons Node id: 5906collectionThis page is under construction
Last Updated May 8, 2023
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[NOTES/CM-ALL] Classical Mechanics --- Repository of Notes for LecturesNode id: 6102collectionThis repository contains collection of NOTES for LECTURES in Classical Mechanics. These are arranged according to topics in the subject. Not sorted or arranged in any particular order within a topic.
Suitable for teachers and content developers only.
Click on any topic to browse all available notes.
(Click opens an embedded window) |
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[LSN/QFT-ALL]: Quantum Field Theory --- Stockpile of Lessons Node id: 3836collection
- Module I :: Classical Field Theory
- Module II:: Second Quantization of Schrodinger Field
- Module III:: Preparation From Quantum Mechanics-I:: Time Dependent Perturbation Theory
- Module IV:: Computation of Cross Sections and Life Times-I
- Module V:: Preparation from Relativistic Quantum Mechanics-I --- Klein Gordon Equation
- Module VI:: Quantization of Klein Gordon Field
- Module VII:: Quantization of Electromagnetic Field
- Module VIII:: Preparation from Relativistic Quantum Mechanics-II ---- Dirac Equation
- Module IX:: Quantization of Dirac equation
- Module-X:: Symmetries and Conservation Laws
- Module-XI:: S- Matrix, Schwinger Dyson Expansion, and Wick's Theorem
- Module-XII:: Feynman Diagrams and Feynman Rules
- Module- XIII:: Computation of Processes --- Tree Diagrams
- Module-XIV:: Higher Order Calculation
- Module-XV:: Anomalous Magnetic Moment of Electron
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[NOTES/QFT-ALL] Quantum Field Theory --- Repository of Notes For LecturesNode id: 6209collectionThis is a repository of NOTES for Lectures in Quantum Field Theory These are arranged according to topics in the subject. Not sorted, or arranged, in any particular order within a topic.
The primary usage of these NOTES is to use them as building blocks for other study resources.
Suitable for teachers and content developers only.
Click on any topic below, to browse all available notes. (Click opens an embedded window) |
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