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inertia-1 transcription, wording and images edits #22

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Sep 6, 2024
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inertia-1 transcription, wording and images edits #22

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1a12cd1
inertia-1 transcription, wording and images edits
Joosty Sep 6, 2024
9edfe90
Merge branch 'angadhn:master' into Inertia-1
Joosty Sep 6, 2024
90db7f0
Merge branch 'angadhn:master' into Inertia-1
Joosty Sep 6, 2024
15cc361
fixes to 'P's
Joosty Sep 6, 2024
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148 changes: 73 additions & 75 deletions inertia/inertia-1.ipynb
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Original file line number Diff line number Diff line change
Expand Up @@ -13,7 +13,7 @@
"id": "0a504f18",
"metadata": {},
"source": [
"- __Mass__: measure of amount of materia in a body. Units of measurement: $kg$, $lb$.\n",
"- __Mass__: the quanity of matter within a given body. The units of measurement are: $kg$, $lb$.\n",
"- __Mass center__: consider a set of particles as shown below which together make up the system of particles $S$:\n",
"\n",
"The $i^{\\text{th}}$ particle has a mass $m_i$\n",
Expand All @@ -25,6 +25,7 @@
"```\n",
"\n",
"$S^*$ is a fictitious particle such that:\n",
"\n",
"<!--\n",
"```{figure} ./images/2.png\n",
"---\n",
Expand All @@ -34,12 +35,11 @@
"```\n",
"-->\n",
"\n",
"```{math}\n",
":label: masscenter_1\n",
"\\sum_{i} m_i {\\bf p}_i = 0\n",
"```\n",
"\n",
"This fictitious particle is called __mass center__. So, how does one locate the mass centre from a point $O$?\n",
"$$\\sum_{i} m_i {\\bf p}_i = 0 \\tag{8.1}$$\n",
"\n",
"\n",
"This fictitious particle is called __mass center__. The mass center can be located by introducing a indipendant point seperate to the body. \n",
"\n",
"```{figure} ./images/3.png\n",
"---\n",
Expand All @@ -56,13 +56,19 @@
"{\\bf p}_n = {\\bf r}^* - {\\bf q}_n\n",
"$$\n",
"\n",
"Thus, expanding {eq}`masscenter_1`\n",
"Thus, expanding ___8.1___:\n",
"\n",
"```{figure} ./images/4.png\n",
"---\n",
"name: 4\n",
"---\n",
"```\n",
"$$\\sum_i {m_i}{\\mathbf{p}_i} =0$$\n",
"\n",
"$$\\rightarrow {m_1}{\\mathbf{p}_1} + {m_2}{\\mathbf{p}_2} + ... + {m_n}{\\mathbf{p}_n} = 0$$\n",
"\n",
"$$\\rightarrow {m_1}{{\\mathbf r}^* - {\\mathbf q}_1} + {m_2}{{\\mathbf r}^* - {\\mathbf q}_2} + ... + {m_i}{{\\mathbf r}^* - {\\mathbf q}_n} = 0$$\n",
"\n",
"$$\\rightarrow {\\mathbf r}^* (m_1 + m_2 + ... + m_n ) = {m_1}{\\mathbf{q}_1} + {m_2}{\\mathbf{q}_2} + ... + {m_n}{\\mathbf{q}_n}$$\n",
"\n",
"$$\\rightarrow {\\mathbf r}^* = \\frac{{m_1}{\\mathbf{q}_1} + {m_2}{\\mathbf{q}_2} + ... + {m_n}{\\mathbf{q}_n}}{(m_1 + m_2 + ... + m_n )}$$\n",
"\n",
"$${\\mathbf r}^* = \\frac{\\sum_i {m_i}{\\mathbf{q}_i}}{\\sum_i {m_i}} \\tag{8.2}$$\n",
"\n",
"For a continuum:\n",
"\n",
Expand All @@ -71,14 +77,9 @@
"name: 5\n",
"---\n",
"```\n",
"$${\\mathbf r}^* = \\frac{\\int {\\mathbf r}^* \\mathrm{d}m}{\\int \\mathrm{d}m} = \\frac{\\int e {\\mathbf r}^* \\mathrm{d}V}{\\int e \\mathrm{d}V} \\tag {8.3}$$\n",
"\n",
"```{figure} ./images/6.png\n",
"---\n",
"name: 6\n",
"---\n",
"```\n",
"\n",
"where, $dm$ is elemental mass that can be obtained from density $\\rho$ and elementar volume $dV$."
"where, $\\mathrm{d}m$ is elemental mass that can be obtained from density $\\rho$ and elementary volume $\\mathrm{d}V$."
]
},
{
Expand All @@ -100,11 +101,7 @@
"---\n",
"```\n",
"\n",
"```{figure} ./images/8.png\n",
"---\n",
"name: 8\n",
"---\n",
"```\n",
"$${\\mathbf r}^* = \\frac{{m_{s_1}{{\\mathbf{r}_1}^*}} + {m_{s_2}{{\\mathbf{r}_2}^*}} + {m_{s_3}{{\\mathbf{r}_3}^*}} + ...}{(m_{s_1} + m_{s_2} + m_{s_3} + ...)} \\tag{8.4}$$\n",
"\n",
"where, \n",
"\n",
Expand Down Expand Up @@ -167,27 +164,21 @@
"\n",
"Then,\n",
"\n",
"```{figure} ./images/10.png\n",
"---\n",
"name: 10\n",
"---\n",
"```\n",
"$$\n",
"X^* = \\frac{{m_{F_1}} {\\frac{a}{2}} + {m_{F_2}} {\\frac{b}{2}} + m_R (0) + m(0)}{m_{F_1} + m_{F_2} + m_R + m}\n",
"$$\n",
"\n",
"Similarly,\n",
"\n",
"```{figure} ./images/11.png\n",
"---\n",
"name: 11\n",
"---\n",
"```\n",
"$$\n",
"Y^* = \\frac{m_{F_1} \\left( a + \\frac{H}{2} \\right) + m_{F_2} \\frac{a}{2}}{m_{F_1} + m_{F_2} + m_R + m}\n",
"$$\n",
"\n",
"and\n",
"\n",
"```{figure} ./images/12.png\n",
"---\n",
"name: 12\n",
"---\n",
"```"
"$$\n",
"Z^* = \\frac{m_R \\frac{L}{2} + m L_1}{m_{F_1} + m_{F_2} + m_R + m}\n",
"$$"
]
},
{
Expand Down Expand Up @@ -229,30 +220,25 @@
"\n",
"$I^{P/O}_{ab}$ is the product of inertia of $P$ along two lines through point $O$ that are parallel to unit vectors $\\hat{n}_a$ and $\\hat{n}_b$.\n",
"\n",
"```{figure} ./images/14.png\n",
"---\n",
"name: 14\n",
"---\n",
"```\n",
"$$\n",
"I^{P/O}_{ab} \\triangleq m \\left( {\\mathbf p} \\times \\hat{n}_a \\right) \\cdot \\left( {\\mathbf p} \\times \\hat{n}_b \\right) \\tag{8.5}\n",
"$$\n",
"\n",
"\n",
"__8.5__ can be extended for both systems of particles and continua.\n",
"\n",
"\n",
"#### Product of inertia of system particles\n",
"\n",
"```{figure} ./images/15.png\n",
"---\n",
"name: 15\n",
"---\n",
"```\n",
"$$\n",
"I^{S/O}_{ab} \\triangleq \\sum_i m_i \\left( {\\mathbf p}_i \\times \\hat{n}_a \\right) \\cdot \\left( {\\mathbf p}_i \\times \\hat{n}_b \\right) \\tag{8.6}\n",
"$$\n",
"\n",
"#### Product of inertia of continua\n",
"\n",
"```{figure} ./images/16.png\n",
"---\n",
"name: 16\n",
"---\n",
"```\n",
"$$\n",
"I^{B/O}_{ab} \\triangleq \\int \\mathrm{d}m \\left( {\\mathbf p} \\times \\hat{n}_a \\right) \\cdot \\left( {\\mathbf p} \\times \\hat{n}_b \\right) \\tag{8.7}\n",
"$$\n",
"\n",
"```{figure} ./images/17.png\n",
"---\n",
Expand Down Expand Up @@ -288,28 +274,36 @@
"\n",
"$I^{P/O}_{aa}$ is the moment of inertia of P about a line through point $O$ which is parallel to the unit vector $\\hat{n}_a$.\n",
"\n",
"```{figure} ./images/19.png\n",
"---\n",
"name: 19\n",
"---\n",
"```\n",
"$$\n",
"I^{P/O}_{a} \\triangleq m \\left( {\\mathbf p} \\times \\hat{n}_a \\right) \\cdot \\left( {\\mathbf p} \\times \\hat{n}_a \\right) \\tag{8.8}\n",
"$$\n",
"\n",
"$$ I^{P/O}_{aa} \\triangleq m \\cdot |{\\mathbf p} \\times \\hat{n}_a|^2$$\n",
"\n",
"$$\n",
"= m \\left( |\\mathbf{p}| \\cdot |\\hat{n}_a| \\cdot \\sin\\theta \\right)^2\n",
"$$\n",
"\n",
"$$\n",
"= m \\left( |\\mathbf{p}| \\cdot 1 \\cdot \\sin\\theta \\right)^2\n",
"$$\n",
"\n",
"\n",
"$$= m d^2$$\n",
"\n",
"__8.8__ can be extended to both systems of particles and continua.\n",
"\n",
"#### Moment of inertia of system of particles\n",
"\n",
"```{figure} ./images/20.png\n",
"---\n",
"name: 20\n",
"---\n",
"```\n",
"#### Moment of inertia of system of particles\n",
"$$\n",
"I^{S/O}_{a} \\triangleq \\sum_i m_i \\left( {\\mathbf p}_i \\times \\hat{n}_a \\right) \\cdot \\left( {\\mathbf p}_i \\times \\hat{n}_a \\right) \\tag{8.9}\n",
"$$\n",
"\n",
"```{figure} ./images/21.png\n",
"---\n",
"name: 21\n",
"---\n",
"```\n",
"#### Moment of inertia of a continua\n",
"\n",
"$$\n",
"I^{B/O}_{a} \\triangleq \\int \\mathrm{d}m \\left( {\\mathbf p}_i \\times \\hat{n}_a \\right) \\cdot \\left( {\\mathbf p}_i \\times \\hat{n}_a \\right) \\tag{8.10}\n",
"$$\n",
"\n",
"```{figure} ./images/22.png\n",
"---\n",
Expand Down Expand Up @@ -371,11 +365,15 @@
"* The off-diagonal elements are the products of inertia.\n",
"* So, the inertia matrix is represented as:\n",
"\n",
"```{figure} ./images/25.png\n",
"---\n",
"name: 25\n",
"---\n",
"```\n",
"$$\n",
"\\left[{I}\\right]^{S/O} =\n",
"\\begin{bmatrix}\n",
"{I^{S/O}_{xx}} & {I^{S/O}_{xy}} & {I^{S/O}_{xz}} \\\\\n",
"{I^{S/O}_{yx}} & {I^{S/O}_{yy}} & {I^{S/O}_{yz}} \\\\\n",
"{I^{S/O}_{zx}} & {I^{S/O}_{zy}} & {I^{S/O}_{zz}} \\\\\n",
"\\end{bmatrix}\n",
"$$\n",
"\n",
"\n",
"```{warning}\n",
"In all cases, $I_{ab} = I_{ba}$ because the formula relies on the dot product of vectors.\n",
Expand Down
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