{
"cells": [
{
"cell_type": "markdown",
"metadata": {
"colab_type": "text",
"id": "view-in-github"
},
"source": [
"![\"Open](\"https://colab.research.google.com/assets/colab-badge.svg\")
"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "69JpzZ3Wz3yc"
},
"source": [
"# Matriz de inercia de un radiómetro de Crookes\n",
"\n",
"Nuestro radiómetro (molinillo de luz) consiste en cuatro aspas de mica, unidas por alambres ideales (sin masa), que pueden pivotar en torno a un punto de apoyo. Idealmente, el molino gira en tornoa un eje vertical, como se muestra en la siguiente figura:\n",
"\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "ZwBQSANh0-sB"
},
"source": [
"Comenzamos calculando los momentos de inercia de cada placa de mica respecto a sus centros de masas, y luego aplicaremos Steiner para trasladar los momentos de inercia al punto de apoyo.\n",
"\n",
"## Momento de inercia de placa rectangular\n",
"\n",
"Cada placa tiene lado $L$ y espesor despreciable. Como ejes principales tomamos dos direcciones en las diagonales (una horizontal, otra vertical) y la dirección perpendicular a cada placa. El eje vertical será $e_3$, los otros dos dependerán de la orientación de las placas.\n",
"\n",
"## - Diagonal vertical\n",
"\n",
"Tomamos elementos longitudinales situados a distancia $x$ del eje, de espesor $dx$, y longitud $h$. Esta longitud está limitada por los bordes rectos de la placa. Por la derecha, la ecuación de la recta que define el borde superior es:\n",
"\n",
"\\begin{equation}\n",
"y_d = \\frac{L}{\\sqrt{2}} - x.\n",
"\\end{equation}\n",
"\n",
"Podemos calcular la longitud del elemento diferencial como $h_d=2y$. Por el lado de la izquierda, el borde superior sigue la ecuación,\n",
"\n",
"\\begin{equation}\n",
"y_i = \\frac{L}{\\sqrt{2}} + x.\n",
"\\end{equation}\n",
"\n",
"Y similarmente calculamos su altura como $h_d=2y$. Así que el momento de inercia se puede calcular como:\n",
"\n",
"\\begin{equation}\n",
"I_3 = \\int_{-\\frac{L}{\\sqrt{2}}}^{0} \\sigma x^2 h_i \\, dx + \\int_{0}^{\\frac{L}{\\sqrt{2}}} \\sigma x^2h_d \\, dx = \\int_{-\\frac{L}{\\sqrt{2}}}^{0} \\sigma x^2 2(\\frac{L}{\\sqrt{2}} + x) \\, dx + \\int_{0}^{\\frac{L}{\\sqrt{2}}} \\sigma x^2 2(\\frac{L}{\\sqrt{2}} - x) \\, dx.\n",
"\\end{equation}\n",
"\n",
"Hacemos ahora las cuentas en SymPy."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"id": "A641SICY42Fk"
},
"outputs": [],
"source": [
"import sympy as smp\n",
"from sympy import latex # para mostrar los resultados\n",
"from IPython.display import Math # para mostrar los resultados"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"id": "QtsCq70n43bT"
},
"outputs": [],
"source": [
"x, y, z = smp.symbols('x y z') # cartesianas\n",
"\n",
"M, L, sigma = smp.symbols('M L sigma') # Masa, lado y densidad superficial de la placa cuadrada.\n",
"h, d = smp.symbols('h, d') #dimensiones del molino"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 54
},
"id": "tp7yJSZgDi0k",
"outputId": "165ddcb1-ee8c-45b4-b6e7-fd874bd07971"
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"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\frac{L^{2} M}{12}$"
],
"text/plain": [
"L**2*M/12"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"D = M/L**2\n",
"\n",
"Iv_cdm = smp.integrate(D*(x**2)*2*((L/smp.sqrt(2) + x)), (x, -L/smp.sqrt(2), 0)) + smp.integrate(D*(x**2)*2*((L/smp.sqrt(2) - x)), (x, 0, L/smp.sqrt(2)))\n",
"Iv_cdm"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "cT3bkqvFEgPS"
},
"source": [
"### Diagonal horizontal\n",
"\n",
"Lo llamamos, por ejemplo, $I_h$ en esta dirección. Por simetría, $I_h$ = $I_v$."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"id": "NvmL1ty9Eu4S"
},
"outputs": [],
"source": [
"Ih_cdm = Iv_cdm"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "ewkAMZl5E9m5"
},
"source": [
"### Dirección perpendicular\n",
"\n",
"Lo llamamos, de momento $I_p$. Como es un cuerpo plano, $I_p = I_h + I_v$."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"id": "VOwFC1_kFJi5"
},
"outputs": [],
"source": [
"Ip_cdm = Ih_cdm + Iv_cdm"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "zI6MpMlfFQRg"
},
"source": [
"### Matriz de inercia $I^*$ para cada placa\n",
"\n",
"Construímos la matriz, teniendo en cuenta que para cada placa habrá que orientar los ejes $I_1$ e $I_2$ correctamente. Numeramos nuestras placas como A, B, C y D, y tenemos:"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 88
},
"id": "winNFLkGFpmg",
"outputId": "e1b5205d-493e-4a6f-fa4f-e971fd5cd81a"
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"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}\\frac{L^{2} M}{6} & 0 & 0\\\\0 & \\frac{L^{2} M}{12} & 0\\\\0 & 0 & \\frac{L^{2} M}{12}\\end{matrix}\\right]$"
],
"text/plain": [
"Matrix([\n",
"[L**2*M/6, 0, 0],\n",
"[ 0, L**2*M/12, 0],\n",
"[ 0, 0, L**2*M/12]])"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"IA_cdm = smp.Matrix([[Ip_cdm, 0, 0], [0, Ih_cdm, 0], [0, 0, Iv_cdm]])\n",
"IA_cdm"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 88
},
"id": "4lC12EUPG26G",
"outputId": "de7dfcfc-ee3b-418f-88f1-a09ca19941f3"
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}\\frac{L^{2} M}{12} & 0 & 0\\\\0 & \\frac{L^{2} M}{6} & 0\\\\0 & 0 & \\frac{L^{2} M}{12}\\end{matrix}\\right]$"
],
"text/plain": [
"Matrix([\n",
"[L**2*M/12, 0, 0],\n",
"[ 0, L**2*M/6, 0],\n",
"[ 0, 0, L**2*M/12]])"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"IB_cdm = smp.Matrix([[Ih_cdm, 0, 0], [0, Ip_cdm, 0], [0, 0, Iv_cdm]])\n",
"IB_cdm"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 88
},
"id": "JYUbFGR6G2rG",
"outputId": "480af757-3617-4362-c0de-c795fd26c058"
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}\\frac{L^{2} M}{6} & 0 & 0\\\\0 & \\frac{L^{2} M}{12} & 0\\\\0 & 0 & \\frac{L^{2} M}{12}\\end{matrix}\\right]$"
],
"text/plain": [
"Matrix([\n",
"[L**2*M/6, 0, 0],\n",
"[ 0, L**2*M/12, 0],\n",
"[ 0, 0, L**2*M/12]])"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"IC_cdm = smp.Matrix([[Ip_cdm, 0, 0], [0, Ih_cdm, 0], [0, 0, Iv_cdm]])\n",
"IC_cdm"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 88
},
"id": "yqH4mlTUG2cG",
"outputId": "1277b928-3d50-41d3-c707-bbb217f432a0"
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}\\frac{L^{2} M}{12} & 0 & 0\\\\0 & \\frac{L^{2} M}{6} & 0\\\\0 & 0 & \\frac{L^{2} M}{12}\\end{matrix}\\right]$"
],
"text/plain": [
"Matrix([\n",
"[L**2*M/12, 0, 0],\n",
"[ 0, L**2*M/6, 0],\n",
"[ 0, 0, L**2*M/12]])"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ID_cdm = smp.Matrix([[Ih_cdm, 0, 0], [0, Ip_cdm, 0], [0, 0, Iv_cdm]])\n",
"ID_cdm"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "7BWtaFhyHQam"
},
"source": [
"Nótese que $I_3$ es igual para todas las matrices. Si el movimiento se da con su velocidad angular contenida exclusivamente en el eje $e_3$, ésta sería la única dirección que nos importaría."
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "af-QYZunkiza"
},
"source": [
"## Steiner\n",
"\n",
"Trasladamos los momentos de cada placa al punto de apoyo, utilizando Steiner:\n",
"\n",
"$\\vec{\\mathbb{I}}_O = \\vec{\\mathbb{I}}_R + \\vec{\\mathbb{I}}^*$\n",
"\n",
"Para calcular \\vec{\\mathbb{I}}_R necesitamos la posición del centro de masas desde $O$, que habrá que calcularlo para cada placa."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 88
},
"id": "UcgeaEiDlrGP",
"outputId": "7c6ea33d-f9e4-4a28-97a8-8ed455212fea"
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\mathbf{I_O} = \\left[\\begin{matrix}\\frac{L^{2} M}{2} + 2 M h^{2} + 2 M \\left(d^{2} + h^{2}\\right) & 0 & 0\\\\0 & \\frac{L^{2} M}{2} + 2 M h^{2} + 2 M \\left(d^{2} + h^{2}\\right) & 0\\\\0 & 0 & \\frac{L^{2} M}{3} + 4 M d^{2}\\end{matrix}\\right]$"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"R_A = smp.Matrix([0, d, h])\n",
"R_B = smp.Matrix([-d, 0, h])\n",
"R_C = smp.Matrix([0, -d, h])\n",
"R_D = smp.Matrix([d, 0, h])\n",
"\n",
"IR_A = M * smp.Matrix([[R_A[1]**2 + R_A[2]**2, -R_A[0]*R_A[1], -R_A[0]*R_A[2]], [-R_A[1]*R_A[0], R_A[0]**2 + R_A[2]**2, -R_A[1]*R_A[2]], [-R_A[2]*R_A[0], -R_A[2]*R_A[1], R_A[0]**2 + R_A[1]**2]])\n",
"IR_B = M * smp.Matrix([[R_B[1]**2 + R_B[2]**2, -R_B[0]*R_B[1], -R_B[0]*R_B[2]], [-R_B[1]*R_B[0], R_B[0]**2 + R_B[2]**2, -R_B[1]*R_B[2]], [-R_B[2]*R_B[0], -R_B[2]*R_B[1], R_B[0]**2 + R_B[1]**2]])\n",
"IR_C = M * smp.Matrix([[R_C[1]**2 + R_C[2]**2, -R_C[0]*R_C[1], -R_C[0]*R_C[2]], [-R_C[1]*R_C[0], R_C[0]**2 + R_C[2]**2, -R_C[1]*R_C[2]], [-R_C[2]*R_C[0], -R_C[2]*R_C[1], R_C[0]**2 + R_C[1]**2]])\n",
"IR_D = M * smp.Matrix([[R_D[1]**2 + R_D[2]**2, -R_D[0]*R_D[1], -R_D[0]*R_D[2]], [-R_D[1]*R_D[0], R_D[0]**2 + R_D[2]**2, -R_D[1]*R_D[2]], [-R_D[2]*R_D[0], -R_D[2]*R_D[1], R_D[0]**2 + R_D[1]**2]])\n",
"\n",
"IO_A = IR_A + IA_cdm\n",
"IO_B = IR_B + IB_cdm\n",
"IO_C = IR_C + IC_cdm\n",
"IO_D = IR_D + ID_cdm\n",
"\n",
"IO = IO_A + IO_B + IO_C + IO_D\n",
"display(Math(r\" \\mathbf{I_O} = \" + smp.latex(IO)))"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "2g5wUzEGzcrU"
},
"source": [
"## b) Sustituimos las dimensiones del cuerpo\n",
"\n",
"- $𝐿 = 16$ mm\n",
"- $𝑑 = 15$ mm\n",
"- $ℎ = 5$ mm\n",
"- $𝜎 = 2.8 g/cm^2$"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 78
},
"id": "_XEaj-Tfzh6s",
"outputId": "3d09d1d1-76d1-4412-c007-e959f7b148d0"
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\mathbf{I_O} = \\left[\\begin{matrix}4.859904 \\cdot 10^{-6} & 0 & 0\\\\0 & 4.859904 \\cdot 10^{-6} & 0\\\\0 & 0 & 7.06286933333333 \\cdot 10^{-6}\\end{matrix}\\right]kg·m^2$"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"\n",
"\n",
"# Sustituimos los valores numéricos\n",
"masa = 28*(16e-3)**2 # en kg\n",
"valores = {L: 16e-3, d: 15e-3, h: 5e-3, M: masa} # en kg y metros\n",
"\n",
"IO_real = IO.subs(valores)\n",
"display(Math(r\" \\mathbf{I_O} = \" + smp.latex(IO_real) + \"kg·m^2\"))"
]
}
],
"metadata": {
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