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Date  |
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[NOTES/CM-09001] Degrees of Freedom of a Rigid BodyBy 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 |
[NOTES/CM-09005] Heavy Symmetrical Top with One Point Fixed$\newcommand{\Label}[1]{\label{#1}}\newcommand{\eqRef}[1]{\eqref{#1}}\newcommand{\Prime}{{^\prime}}\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}} \newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}}\newcommand{\dd}[2][]{\frac{d#1}{d #2}}$ We set up Lagrangian for a heavy symmetrical top and show that the solution can be reduced to quadratures. |
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24-05-14 17:05:13 |
[NOTES/CM-09004] Kinetic Energy of Rigid BodyAn 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} |
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24-05-14 05:05:01 |
[NOTES/CM-08010] Motion in Frames with Linear Acceleration$\newcommand{\Prime}{{^\prime}}\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}} \newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}} \newcommand{\dd}[2][]{\frac{d#1}{d #2}}\newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}}$ 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 |
[NOTES/CM-08008] Proper Rotations and $SO(3)$$\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 |
[NOTES/CM-08012] Matrices for Rotations about Coordinate AxesThe rotation matrices for rotations about the three axes are listed. |
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24-05-03 08:05:13 |
[NOTES/CM-08005] Finite Rotations of Vectors about an Arbitrary Axis$\newcommand{\Prime}{{^\prime}}$ 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 |
[NOTES/CM/08001] The Group of Orthogonal Matrices in Three DimensionsThe groups of all orthogonal matrices is defined It has a subgroup of matrices with determinant +1, This subgroup is called specail orthogonal group. |
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24-05-01 06:05:13 |
[NOTES/CM/The Group of Special orthogonal Matrices Three Dimensions] |
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24-04-28 06:04:46 |
[NOTES/TH-09001] Postulates of Thermodynamics |
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24-04-24 04:04:13 |