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Aparentemente, todos los objetos celestes giran en torno a la [estrella polar](https://es.wikipedia.org/wiki/Estrella_polar) a lo largo de la noche. -VÍDEO +{{< figure library="true" src="interaccion-gravitatoria-2Bach-fisica/movimiento-aparente.jpg" title="https://depositphotos.com/photo/bright-traces-stars-revolving-polar-star-form-circular-tracks-glow-201868948.html" >}} Si observáis alguna estrella que no titila, se trata de un [planeta](https://es.wikipedia.org/wiki/Planeta), del griego πλανήτης (errante). Los que más brillan son [Venus]() y [Júpiter]() (con prismáticos pueden verse incluso 4 de sus [lunas](https://es.wikipedia.org/wiki/Satélites_de_Júpiter)). Se ven mejor al amanecer o al atardecer, de ahí el nombre [_lucero del alba_](https://es.wikipedia.org/wiki/Lucero_del_alba) o _estrella de la mañana_ para Venus. Los planetas que se observan se sitúan sobre una misma línea, conocida como la [eclíptica](https://es.wikipedia.org/wiki/Eclíptica), y las constelaciones que pasan por la eclíptica son las del [zodiaco](https://es.wikipedia.org/wiki/Zodiaco). {{< /spoiler >}} @@ -87,7 +87,7 @@ No se suele mencionar pero fue el [**primer modelo heliocéntrico**](https://es. - [**Paralaje**](https://es.wikipedia.org/wiki/Paralaje) de las **estrellas fijas**: deberían verse en diferente posición dependiendo de la época del año. Aristarco lo justificó de forma esencialmente correcta: dado que la distancia a las estrellas es mucho mayor que el diámetro de la órbita terrestre, no se aprecia paralaje. -- **Implicaciones mecánicas** del movimiento de **rotación terrestre**. Si arrojamos una piedra desde una torre, no debería caer en su base, sino a cierta distancia, pero las observaciones no se correspondían. Esto era una afirmación claramente errónea, pero para comprenderlo se necesitaba conocer el [principio de inercia (1ª ley de Newton)]({{< relref "apuntes-cuarto-eso/fuerzas/index.md#1ª-ley-ley-de-la-inercia" >}}) desconocido para los griegos. +- **Implicaciones mecánicas** del movimiento de **rotación terrestre**. Si arrojamos una piedra desde una torre, no debería caer en su base, sino a cierta distancia, pero las observaciones no se correspondían. Esto era una afirmación claramente errónea, pero para comprenderlo se necesitaba conocer el principio de inercia de Galileo (lo que más tarde sería la [1ª ley de Newton]({{< relref "apuntes-cuarto-eso/fuerzas/index.md#1ª-ley-ley-de-la-inercia" >}})), desconocido para los griegos. ### Ptolomeo (100 d. C.) @@ -122,6 +122,10 @@ Perfeccionó el [telescopio](https://es.wikipedia.org/wiki/Telescopio) y esto pe Estableció las [tres leyes](#leyes-de-kepler) que llevan su nombre, con las que fue de capaz de explicar a la perfección las observaciones. +En este **vídeo**, [**Timothy Rowe**](https://www.youtube.com/@_TimothyRowe/) nos explica (en inglés) los fundamentos y problemas de la astronomía ptolemaica, así como la propuesta de Copérnico (basada en la de Aristarco) y la ayuda de Kepler (órbitas elípticas): + +{{< youtube F3Ycj1VbB_k >}} + ## Leyes de Kepler Constituyen una descripción cinemática del Sistema Solar y consiguen explicar de forma muy precisa todas las observaciones relativas no solo al movimiento de planetas, sino también [asteroides](https://es.wikipedia.org/wiki/Asteroide), [cometas](https://es.wikipedia.org/wiki/Cometa) y [satélites](https://es.wikipedia.org/wiki/Satélite_natural) (incluso [artificales](https://es.wikipedia.org/wiki/Satélite_artificial)). @@ -219,18 +223,22 @@ $$ Una magnitud se conserva cuando permanece constante, es decir, no cambia con el tiempo. Estudiemos la variación de $\vec L$ con el tiempo para ver en qué circunstancias se conserva: +{{< math >}} $$ \frac{\mathrm d \vec L}{\mathrm d t} = \frac{\mathrm d}{\mathrm d t}\left(\vec r\times m\vec v\right) = \frac{\mathrm d \vec r}{\mathrm d t}\times m\vec v + \vec r\times m\frac{\mathrm d \vec v}{\mathrm d t} = \cancelto{0}{\vec v \times m \vec v} + \vec r\times m\vec a = \vec r\times \vec F $$ +{{< /math >}} $\vec L$ se conserva si su variación con el tiempo es cero: +{{< math >}} $$ \frac{\mathrm d \vec L}{\mathrm d t} = \vec r\times \vec F = 0 \Leftrightarrow \begin{cases} \vec F = 0 & \text{La fuerza neta que actúa sobre el cuerpo es cero.} \\ \vec r\parallel \vec F & \text{Sucede en fuerzas centrales.} \end{cases} $$ +{{< /math >}} ### Movimientos planetarios @@ -268,7 +276,7 @@ La conservación de $\vec L$ también explica la [2ª ley de Kepler](#2ª-ley-le La ley de gravitación universal introdujo la idea de [acción a distancia](https://es.wikipedia.org/wiki/Acción_a_distancia). ¿Cómo puede un objeto influenciar a otro sin estar en contacto con él? Hubo varios intentos de explicar cómo se producía la interacción a distancia. Algunos científicos afirmaban que existía una sustancia llamada [éter]() que llenaba el vacío (_fluido_) y era la responsable de transmitir la interacción. Este éter no se sabía de qué estaba hecho y su existencia no podía demostrarse experimentalmente, pero se mantuvo durante algún tiempo. -Científico posteriores demostraron que la interacción a distancia se producía también en el vacío, por lo que la existencia del éter no era necesaria. Finalmente, [Einstein](https://es.wikipedia.org/wiki/Albert_Einstein) justificó la acción a distancia por una perturbación en el [espacio-tiempo](https://es.wikipedia.org/wiki/Espacio-tiempo) debida a la presencia de una masa. Dicha perturbación se denominó [**campo gravitatorio**](https://es.wikipedia.org/wiki/Campo_gravitatorio). +Científicos posteriores demostraron que la interacción a distancia se producía también en el vacío, por lo que la existencia del éter no era necesaria. Finalmente, [Einstein](https://es.wikipedia.org/wiki/Albert_Einstein) justificó la acción a distancia por una perturbación en el [espacio-tiempo](https://es.wikipedia.org/wiki/Espacio-tiempo) debida a la presencia de una masa. Dicha perturbación se denominó [**campo gravitatorio**](https://es.wikipedia.org/wiki/Campo_gravitatorio). {{< figure library="true" src="interaccion-gravitatoria-2Bach-fisica/espacio-tiempo.jpg" title="Representación artística que visualiza el espacio-tiempo como una superficie bidimensional simplificada, que está siendo distorsionada por la presencia de tres cuerpos masivos, representados como esferas de colores. La distorsión causada por cada esfera es proporcional a su masa. Fuente: https://www.esa.int/ESA_Multimedia/Images/2015/09/Spacetime_curvature." >}} @@ -303,9 +311,11 @@ expresión que coincide con la [**ley de gravitación universal de Newton**](#le ### Potencial gravitatorio Como la **fuerza gravitatoria** es [**conservativa**](https://es.wikipedia.org/wiki/Fuerza_conservativa), el campo gravitatorio $\vec g$ puede expresarse en función de un [potencial gravitatorio](https://es.wikipedia.org/wiki/Potencial_gravitatorio) $V$, definido como el trabajo por unidad de masa que debe realizar una fuerza para traer un cuerpo desde el infinito (donde $V=0$ por convenio) hasta un punto P, a velocidad constante: +{{< math >}} \[ V = \frac{W_{\infty\rightarrow P}}{m} = \frac{1}{m}\int_\infty^P \vec F\cdot \mathrm d \vec r = \frac{1}{m}\int_\infty^P \frac{GMm}{r^2}\,\mathrm d r = -\frac{GM}{r} \] +{{< /math >}} tal que $\vec g = -\nabla V$. #### Distribución de masas puntuales @@ -316,10 +326,12 @@ V_\mathrm t = \sum_{i=1}^n V_i = \sum_{i=1}^n \left(-\frac{GM_i}{r_i}\right) ### Energía potencial gravitatoria -La energía potencial gravitatoria, $E_\mathrm p$, es la energía que adquiere una masa $m$ en el seno de un campo gravitatorio $\vec g$. Se define como el trabajo que debe realizar una fuerza para traer un cuerpo desde el infinito (donde $V=0$ por convenio) hasta un punto P, a velocidad constante: +La energía potencial gravitatoria, $E_\mathrm p$, es la energía que adquiere una masa $m$ dentro de un campo gravitatorio $\vec g$. Se define como el trabajo que debe realizar una fuerza para traer un cuerpo desde el infinito (donde $V=0$ por convenio) hasta un punto P, a velocidad constante: +{{< math >}} \[ E_\mathrm p = W_{\infty\rightarrow P} = \int_\infty^P \vec F\cdot \mathrm d \vec r = \int_\infty^P \frac{GMm}{r^2}\,\mathrm d r = -\frac{GMm}{r} \] +{{< /math >}} También se puede calcular a partir del potencial gravitatorio: \[ @@ -330,13 +342,16 @@ El signo $-$ indica que la masa está ligada al campo gravitatorio (si $E_\mathr ### Trabajo realizado por la fuerza gravitatoria Cuando una masa $m$ se mueve libremente dentro de un campo gravitatorio, se realiza un trabajo $W_{1\rightarrow 2}$ tal que: +{{< math >}} \[ \begin{aligned} W_{1\rightarrow 2} = \int_1^2 \vec F\cdot \mathrm d \vec r = \int_1^2 -F\,\mathrm d r &= \int_1^2 -\frac{GMm}{r^2}\,\mathrm d r \\ &= \left. GMm\frac{1}{r}\right |_1^2 = \frac{GMm}{r_2} - \frac{GMm}{r_1} =-\symup\Delta E_\mathrm p = -m\symup\Delta V, \end{aligned} \] +{{< /math >}} que solo depende de las posiciones inicial y final, lo que significa que $\vec F_\mathrm g$ es **conservativa**. +{{< math >}} \[ \begin{cases} W=0 & \text{La masa $m$ describe una trayectoria cerrada (o se mueve por una superficie equipotencial).} \\ @@ -344,6 +359,7 @@ W>0 & \text{La masa $m$ se acerca a $M$, desplazándose hacia $V$ decrecientes ( W<0 & \text{La masa $m$ se aleja de $M$ (proceso no espontáneo, se requiere una $\vec F_\mathrm{externa}$).} \end{cases} \] +{{< /math >}} ### Representación del campo gravitatorio @@ -389,26 +405,26 @@ Una esfera que solo tiene masa en su superficie: Una esfera con densidad homogénea $\rho$: - Para $r \leq R$: - $$ + {{< math >}} \left.\begin{aligned} \Phi &= \oint_S \vec g \cdot \mathrm d \vec s = -g\oint_S \mathrm d s = -g\cdot 4\pi r^2 \\ \Phi &= -4\pi G\cdot M_\mathrm{encerrada} = -4\pi G\rho\cdot\frac{4}{3}\pi r^3 \end{aligned}\right\rbrace \vec g = -\dfrac{4\pi G\rho}{3}r\,\hat{\mathrm u}_\mathrm r\ (\text{proporcional al radio $r$}) - $$ + {{< /math >}} - Para $r \geq R$: - $$ + {{< math >}} \left.\begin{aligned} \Phi &= \oint_S \vec g \cdot \mathrm d \vec s = -g\oint_S \mathrm d s = -g\cdot 4\pi r^2 \\ \Phi &= -4\pi G\cdot M \end{aligned}\right\rbrace \vec g = -\frac{GM}{r^2}\,\hat{\mathrm u}_\mathrm r\ (\text{igual que masa puntual}) - $$ + {{< /math >}} ### Campo gravitatorio terrestre Lo que acabamos de ver para una esfera maciza homogénea también es aplicable a esferas con capas homogéneas concéntricas, por lo que podemos aplicarlo a la Tierra[^4]: $$ -\vec g = -\frac{GM_\mathrm T}{r^2}\,\hat{\mathrm u}_\mathrm r \qquad \mathrm{para}\ r=R_\mathrm T+h\geq R_\mathrm T\ (h\geq 0) +\vec g = -\frac{GM_\mathrm T}{r^2}\,\hat{\mathrm u}_\mathrm r \quad \mathrm{para}\ r=R_\mathrm T+h\geq R_\mathrm T\ (h\geq 0) $$ [^4]: La realidad es un poco más complicada. A día de hoy tenemos el llamado [Modelo de Referencia Preliminar](https://es.wikipedia.org/wiki/Modelo_de_referencia_preliminar_de_la_Tierra) (PREM por sus siglás en inglés): @@ -430,9 +446,11 @@ g_0 = \frac{GM_\mathrm T}{R_\mathrm T^2} $$ A una altura $h$ sobre la superficie: +{{< math >}} $$ g = \frac{GM_\mathrm T}{\left(R_\mathrm T+h\right)^2} = g_0\frac{R_\mathrm T^2}{\left(R_\mathrm T+h\right)^2} = g_0\frac{1}{\left(1+\dfrac{h}{R_\mathrm T}\right)^2} $$ +{{< /math >}} #### Variación de $\vec g$ con la latitud @@ -446,12 +464,14 @@ $$ $$ donde $a_{\mathrm{cf}_\mathrm r} = a_\mathrm{cf}\cos\varphi = \omega^2 r \cos\varphi = \omega^2R_\mathrm T\cos^2\varphi$. El punto P siente una gravedad efectiva inferior ya que la componente radial algo contrarresta: +{{< math >}} $$ g_\mathrm{efectiva} = g-a_{\mathrm{cf}_\mathrm r} = g-\omega^2R_\mathrm T\cos^2\varphi = \begin{cases} - g-\omega^2 R_\mathrm T & \varphi=0°\ (\text{ecuador}) \\ - g & \varphi=90°\ (\text{polos}) + g-\omega^2 R_\mathrm T & \varphi=0°\ (\text{ecuador}) \\ + g & \varphi=90°\ (\text{polos}) \end{cases} $$ +{{< /math >}} {{< figure library="true" src="interaccion-gravitatoria-2Bach-fisica/variacion-g-altura-latitud.svg" title="Variación de la aceleración de la gravedad con la altura y la latitud. Se ve claramente cómo disminuye con la altura y es mínima en el ecuador ($\varphi = 0°$) y máxima en los polos $\varphi = 90°$." width="100%" lightbox="false" theme="light" >}} @@ -490,21 +510,26 @@ de donde podemos obtener: ### Satélites geoestacionarios Son aquellos cuyo periodo coincide con el [periodo de rotación terrestre](https://es.wikipedia.org/wiki/Día), de forma que orbitan manteniéndose siempre en la misma vertical (aparentemente no se mueven). Se colocan en el plano del ecuador para que $\vec g$ no varíe con la latitud. Despejando de la [3ª ley de Kepler](#3ª-ley-ley-de-los-periodos): +{{< math >}} $$ \frac{T^2}{r^3} = \frac{4\pi^2}{GM_\mathrm T} \rightarrow r = \sqrt[3]{\frac{GM_\mathrm T}{4\pi^2}} \rightarrow h = r-\mathrm R_T \approx 35\,800\,\mathrm{km} $$ +{{< /math >}} ### Energía de los satélites Cuando describe una órbita, la energía mecánica de un satélite viene dada por: +{{< math >}} \[ E = E_\mathrm c + E_\mathrm p = \frac{1}{2}mv^2 - \frac{GMm}{r} = \frac{1}{2}\frac{GMm}{r} - \frac{GMm}{r} = -\frac{1}{2}\frac{GMm}{r} = \frac{E_\mathrm p}{2} \] +{{< /math >}} Si tenemos en cuenta que todas las **fuerzas** que actúan son **conservativas**, la **energía mecánica se conserva**. Así, es posible calcular: Velocidad de lanzamiento : Es la velocidad que hay que comunicar a un satélite para colocarlo en su órbita, suponiendo que se lanza desde la superficie terrestre: +{{< math >}} $$ \begin{aligned} E_1 & = E_2 \\ @@ -512,6 +537,7 @@ E_{\mathrm c_1} + E_{\mathrm p_1} & = \frac{E \frac{1}{2}\cancel mv^2 - \frac{GM_\mathrm T \cancel m}{R_\mathrm T} & = -\frac{GM_\mathrm T \cancel m}{2r} \rightarrow v = \sqrt{2GM_\mathrm T\left(\frac{1}{R_\mathrm T}-\frac{1}{2r}\right)} \end{aligned} $$ +{{< /math >}} Velocidad de escape : Si en el lanzamiento proporcionamos al satélite una velocidad suficientemente elevada, este puede llegar a escapar del campo gravitatorio terrestre. A esta (mínima) velocidad se le denomina [**velocidad de escape**](https://es.wikipedia.org/wiki/Velocidad_de_escape). Imponiendo la condición de que el objeto se aleje hasta una distancia infinita y quede en reposo ($E=0$): @@ -536,9 +562,11 @@ Se cree que la mayoría de las galaxias giran en torno a un agujero negro superm Trabajo para pasar de una órbita a otra : Supongamos que queremos pasar un satélite de una órbita 1 a otra órbita 2. Será necesario realizar un trabajo tal que: +{{< math >}} \[ E_1+W = E_2 \rightarrow W = E_2-E_1 = - \frac{GM_\mathrm T m}{2r_2}-\left(- \frac{GM_\mathrm T m}{2r_1}\right) = \frac{GM_\mathrm T m}{2}\left(\frac{1}{r_1}-\frac{1}{r_2}\right) \] +{{< /math >}} {{< figure library="true" src="interaccion-gravitatoria-2Bach-fisica/orbita-hohmann.svg" title="La [**órbita de transferencia de Hohmann**](https://es.wikipedia.org/wiki/Órbita_de_transferencia_de_Hohmann) es una maniobra orbital que traslada a una nave espacial desde una órbita circular a otra utilizando dos impulsos de su motor. El nombre proviene del científico alemán [Walter Hohmann](https://es.wikipedia.org/wiki/Walter_Hohmann) que publicó su teoría en 1925. Adaptada de https://commons.wikimedia.org/wiki/File:TransferOrbit.svg." width="100%" lightbox="false" theme="light" >}} @@ -550,19 +578,23 @@ E = \overbrace{\frac{1}{2}mv^2}^{+}\overbrace{-\frac{GMm}{r}}^{-} \] $E<0$ Ligado al campo (órbitas cerradas): +{{< math >}} $$ \begin{cases} E_\mathrm c = 0 & \text{El satélite cae por atracción gravitatoria.} \\ E_\mathrm c \neq 0 & \text{Describe una órbita cerrada (circular o elíptica). Ej. satélites o planetas.} \end{cases} $$ +{{< /math >}} $E\geq 0$ No ligado al campo (órbitas abiertas): +{{< math >}} \[ \begin{cases} E = 0 & \text{Órbita parabólica: $E_\mathrm c = -E_\mathrm p$. Ej. cometas.} \\ E > 0 & \text{Órbita hiperbólica: $E_\mathrm c > E_\mathrm p$. Ej. asteroides.} \end{cases} \] +{{< /math >}} **Explora** la **relación** entre la **gravedad** y las **órbitas** con estas excelentes **simulaciones**: diff --git a/content/es/apuntes-segundo-bach-fisica/interaccion-gravitatoria/interaccion-gravitatoria-poster.pdf b/content/es/apuntes-segundo-bach-fisica/interaccion-gravitatoria/interaccion-gravitatoria-poster.pdf index a1ae8149e0dd801cb595d33fa58d76b14ab23c7a..c783ddfaaf28f248552a57a070866ab8cf6aeb58 100644 GIT binary patch delta 22905 zcmY(pQ*bU!&@CL>wr%d%wr$(laq`5rZQHhO8#}h0{O@xp6cnTsj2GK-BrD& zd+QN8>k<5$frC;&Sh(6kQh+z$0s7+KjRK1A;b*Zf>eZILh9iTVU^STD_>={NTj z6nK%~j*%$IcTZ|HXxa9TwMN#W+e#cl+DeFXh!8pcD8Xio$u2=G>Y20AmT>GLpv<4K zqH+|087e%GA@nJ5O9Fdl;x1dVt@vk%_eKAx#<6PicVVg6YEVXPGoh400k|ksL2cNn zb9q=DSQ{)G4a_4f2VF5(T^187f_J&{u7@wuwY$y!PTyV+CkAn5E?&^d#ecB3F>TJ$ zm?O|fn_C$*llItaV$Jwsj9UIP&!i>F=o6-!)IgJgJ#uL#+P7{(49SHG{hd!0H$)~9 z8y=YVaNKLxXPpn#U3Dhgy$j;}#OnAOcEV7yATZ{^f8kX5( 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zBem1gBatcdEXc@5ud1kk`@-cd13ghM;9$!z4suW;BK9kU9IP9qdaMG5>sp?{P=iQ^ z?j!*eI}&Dj;Dstd?swO^L#9E3QAcfQZjC(iz%kZ;+o1fA z5#4LEkPC2Z)Hw)+jqL3nw}Ie;Rimtqe@((B5=62rCf-fN^t;4akiy;9!#PyJZ=z5Y z0;r!LwT21fiD;3B#r#kCg028$IKLnt-&N?p{KWh>u3XFczd%VMC-l=oNU!cscaV?Z zBgAKKRZB2kl=3*_2t&~UAwmwA#|Y3C=Pnm7vyMnA6R7_WD-?c;eqr3SJ(*e zaqexSo>UrPjij>-DISUEj84YFdk_Q*)b%822`q_7uQ-oCe+|6=(z=!XNDVGj{FH8q z>K4U6a>LECoB8(M>?sCpo1O%M9uVTkx^wz`!S(6(6Y6~N*UgD=V1RMmG!TV`#9V*$ zL4_bYy2IEYVIaAv$T}4gkQ_qsmnyb$1hHA;Q>h%N3&C>CO~+0p6R-vCiGY(JijFnu zxTxR(lM~kA)&SYx;TAcFk^<@2PFhSt3{z{S6`vAc(zR2y@-vrKh@dAn9=6W|+{9t9EyXOxAIuqIW$^}@r5 zTR*v&Z8M6~j*oY-mJ7^M=?!A*x5g6P_(8Wej`Z$NRaEJP=^<+t5B$ok$wlu921Ev$ zHW#%ZkM33TEXO9^p`gSJ?@8-#*7&K-9o@jp4DyU6^?Gbu%uPm{^y$V>oy(&#@Tp;8 zmGmkQXb0$pk)QdFgq^g6OO5( +--- + +{{% section %}} + +## Modelos cosmológicos + +- [Aristóteles (350 a. C.)](#/1/1) +- [Aristarco de Samos (300 a. C.)](#/1/8) +- [Ptolomeo (100 d. C.)](#/1/13) +- [Copérnico (s. XV)](#/1/16) +- [Galileo (s. XVI)](#/1/21) +- [Kepler (s. XVI–XVII)](#/1/24) + +(continúa hacia abajo) + +👇 + +--- + +### Aristóteles (350 a. C.) + +{{< figure library="true" src="interaccion-gravitatoria-2Bach-fisica/aristoteles.jpg" width="45%" >}} + +--- + +Era un [**modelo geocéntrico**](https://es.wikipedia.org/wiki/Teoría_geocéntrica), con la Tierra en el centro del Universo, rodeada por esferas transparentes de distintos tamaños en las que se encontraban los diferentes planetas. + +--- + +La esfera más externa era en la que se encontraban las estrellas fijas. Dicha esfera giraba a un ritmo constante en torno a un eje que pasaba por la estrella polar. + +--- + +Su movimiento era transmitido a las esferas de planetas más externos por rozamiento y así sucesivamente. Era, por tanto, un **universo mecánico**. + +--- + +{{< figure library="true" src="interaccion-gravitatoria-2Bach-fisica/modelo-aristoteles.jpg" width="70%" >}} + +--- + +Como el mundo celeste era perfecto, los objetos en él solo podían describir órbitas circulares puesto que el círculo era el símbolo de la perfección. + +--- + +Este modelo podía explicar muchas de las observaciones, pero no explicaba, por ejemplo: + +- Variaciones observadas en el brillo del Sol (debidas a cambios en la distancia a la Tierra). +- [Movimientos retrógrados](https://es.wikipedia.org/wiki/Retrogradación_de_los_planetas) observados en algunos planetas que, aparentemente, cambian el sentido de su movimiento. + +--- + +### Aristarco de Samos (300 a. C.) + +{{< figure library="true" src="interaccion-gravitatoria-2Bach-fisica/aristarco.jpg" width="40%" >}} + +--- + +No se suele mencionar pero fue el [**primer modelo heliocéntrico**](https://es.wikipedia.org/wiki/Teoría_heliocéntrica). En su modelo, el Sol se situaba en el centro del Universo y el resto de los planetas giraban en torno a él en órbitas circulares de distintas dimensiones. + +--- + +El modelo de Aristarco también suponía que la Tierra describía un [movimiento de rotación](https://es.wikipedia.org/wiki/Rotación_de_la_Tierra) alrededor de su eje que permitía explicar fácilmente el día y la noche. + +--- + +A pesar de que este modelo era bastante acertado, no prosperó por razones de índole filosófica: + +- [**Paralaje**](https://es.wikipedia.org/wiki/Paralaje) de las **estrellas fijas**: deberían verse en diferente posición dependiendo de la época del año. Aristarco lo justificó de forma esencialmente correcta: dado que la distancia a las estrellas es mucho mayor que el diámetro de la órbita terrestre, no se aprecia paralaje. + +--- + +- **Implicaciones mecánicas** del movimiento de **rotación terrestre**. Si arrojamos una piedra desde una torre, no debería caer en su base, sino a cierta distancia, pero las observaciones no se correspondían. Esto era una afirmación claramente errónea, pero para comprenderlo se necesitaba conocer el principio de inercia de Galileo (lo que más tarde sería la [1ª ley de Newton]({{< relref "apuntes-cuarto-eso/fuerzas/index.md#1ª-ley-ley-de-la-inercia" >}})), desconocido para los griegos. + +--- + +### Ptolomeo (100 d. C.) + +Propuso un modelo también [**geocéntrico**](https://es.wikipedia.org/wiki/Teoría_geocéntrica) introduciendo ciertas variaciones al modelo de Aristóteles, como que los planetas podían describir movimientos circulares o una composición de estos. + +--- + +Para ello, introdujo varios artificios matemáticos, como los [**epiciclos**](https://es.wikipedia.org/wiki/Epiciclo), el **deferente** y el [**ecuante**](https://es.wikipedia.org/wiki/Ecuante). Encajando todos estos artificios, Ptolomeo consiguió un sistema útil para astrónomos y navegantes, que perduró más de 14 siglos. + +--- + +{{< figure library="true" src="interaccion-gravitatoria-2Bach-fisica/modelo-ptolomeo.svg" width="75%" lightbox="false" >}} + +--- + +### Copérnico (s. XV) + +{{< figure library="true" src="interaccion-gravitatoria-2Bach-fisica/copernico.jpg" width="65%" >}} + +--- + +Propuso un [**modelo heliocéntrico**](https://es.wikipedia.org/wiki/Teoría_heliocéntrica) igual que el de [Aristarco](#aristarco-de-samos-300-a-c) (Sol en el centro con planetas orbitando alrededor y rotación de la Tierra). + +--- + +Este modelo explicaba de forma mucho más sencilla los [movimientos retrógrados](https://es.wikipedia.org/wiki/Retrogradación_de_los_planetas), que eran causados por la combinación entre el movimiento terrestre y el de los planetas. + +--- + +Presentaba varias **ventajas** sobre el geocéntrico: + +- Simplicidad: se requieren muchos menos movimientos para explicar las observaciones. +- Modelo único: todos los planetas se ajustan de igual forma al modelo. + +--- + +Sin embargo, tardó más de un siglo en ser aceptado por el rechazo a perder la cosmología aristotélica, los dogmas y porque no existía ninguna observación que un modelo geocéntrico no pudiera explicar. + +--- + +### Galileo (s. XVI) + +> **Aprende más** sobre **Galileo Galilei** echando un vistazo a su **biografía científica** en nuestra sección de [**Historia de la Ciencia**](https://fisiquimicamente.com/recursos-fisica-quimica/historia-ciencia/). + +--- + +Perfeccionó el [telescopio](https://es.wikipedia.org/wiki/Telescopio) y esto permitió mejorar enormemente las observaciones. Así, descubrió las fases de Venus, que únicamente podían ser explicadas de forma satisfactoria considerando un modelo heliocéntrico (se consideró la prueba empírica definitiva). + +--- + +{{< figure library="true" src="interaccion-gravitatoria-2Bach-fisica/telescopio-galileo.jpg" width="50%" >}} + +--- + +### Kepler (s. XVI–XVII) + +Estableció las [tres leyes](#leyes-de-kepler) que llevan su nombre, con las que fue de capaz de explicar a la perfección las observaciones. + +--- + +{{< youtube F3Ycj1VbB_k >}} + +{{% /section %}} + +--- + +{{% section %}} + +## Leyes de Kepler + +- [1ª ley: órbitas elípticas](#/2/2) +- [2ª ley: ley de las áreas](#/2/3) +- [3ª ley: ley de los periodos](#/2/4) + +(continúa hacia abajo) + +👇 + +--- + +Constituyen una descripción cinemática del Sistema Solar y consiguen explicar de forma muy precisa todas las observaciones relativas no solo al movimiento de planetas, sino también [asteroides](https://es.wikipedia.org/wiki/Asteroide), [cometas](https://es.wikipedia.org/wiki/Cometa) y [satélites](https://es.wikipedia.org/wiki/Satélite_natural) (incluso [artificales](https://es.wikipedia.org/wiki/Satélite_artificial)). + +--- + +### 1ª ley: órbitas elípticas + +> Los planetas giran en torno al Sol describiendo **órbitas elípticas**, con el Sol situado en uno de los focos. + +--- + +### 2ª ley: ley de las áreas + +> Los planetas se mueven con [velocidad areolar](https://es.wikipedia.org/wiki/Velocidad_areolar) constante, es decir, el radio vector barre áreas iguales en tiempos iguales. + +$$ +\frac{\mathrm d A}{\mathrm d t} = \text{constante} +$$ + +--- + +### 3ª ley: ley de los periodos + +> El cuadrado del periodo orbital de los planetas es directamente proporcional al cubo de su distancia media al Sol. + +$$ +\frac{T^2}{r^3} = \text{constante} +$$ + +--- + +{{< figure library="true" src="interaccion-gravitatoria-2Bach-fisica/leyes-kepler.svg" width="65%" lightbox="false" >}} + +--- + +**Explora** las **leyes** de **Kepler** con esta magnífica **simulación**: + + + +{{% /section %}} + +--- + +{{% section %}} + +## Ley de gravitación universal de Newton + +Teniendo en cuenta las leyes de Kepler, Newton llegó a la conclusión de que el movimiento de los planetas era debido a una fuerza de atracción proporcional a la masa del planeta ($m$) y del Sol ($M$) e inversamente proporcional al cuadrado de la distancia que los separa ($r$): + +--- + +$$ +\vec F_\mathrm g = -\frac{GMm}{r^2}\,\hat{\mathrm u}_\mathrm r, +$$ + +donde $G = 6.67\times 10^{11}\,\mathrm N \,\mathrm m^2\,\mathrm{kg}^{-2}$ es la [constante de gravitación universal](https://es.wikipedia.org/wiki/Constante_de_gravitación_universal), determinada experimentalmente por [Henry Cavendish](https://es.wikipedia.org/wiki/Henry_Cavendish) en 1798 utilizando una [balanza de torsión](https://es.wikipedia.org/wiki/Balanza_de_torsión). + +--- + +### Medida experimental de $G$ + +La constante de gravitación universal, $G$, fue determinada experimentalmente 100 años después de Newton por el científico británico y francés [Henry Cavendish](https://es.wikipedia.org/wiki/Henry_Cavendish). + +--- + +Para ello, empleó una [balanza de torsión](https://es.wikipedia.org/wiki/Balanza_de_torsión): + +{{< figure library="true" src="interaccion-gravitatoria-2Bach-fisica/experimento-Cavendish.svg" width="55%" lightbox="false" >}} + +--- + +**Experimenta** con este **laboratorio** de **fuerza** de **gravedad**: + + + +--- + +Esta ley no solo es aplicable a cuerpos celestes, sino que también se cumple para dos masas cualesquiera (caída de cuerpos). Se conoce como [**ley de gravitación universal**](https://es.wikipedia.org/wiki/Ley_de_gravitación_universal) y es revolucionaria por muchas razones: + +- Ruptura de la barrera cielo-Tierra. +- Acción a distancia. + +--- + +También es aplicable a un conjunto de [masas puntuales](https://es.wikipedia.org/wiki/Masa_puntual), pues cumple el [**principio de superposición**](https://es.wikipedia.org/wiki/Principio_de_superposición): + +$$ +\vec F_\mathrm t = \sum_i \vec F_i +$$ + +--- + +### Relación con la 3ª ley de Kepler + +En el movimiento de un planeta alrededor del Sol, la fuerza gravitatoria actúa como [fuerza centrípeta](https://es.wikipedia.org/wiki/Fuerza_centrípeta). + +--- + +Suponiendo una órbita circular, podemos relacionar la velocidad orbital con el periodo, $v = 2\pi r/T$: + +$$ +\begin{aligned} +F_\mathrm g & = F_\mathrm c \\ + \frac{GM\cancel m}{r^2} & = \frac{\cancel mv^2}{r} \\ + \frac{GM}{r^2} & = \frac{4\pi^2 r^{\cancel 2}}{T^2 \cancel r} \\ + \frac{T^2}{r^3} & = \frac{4\pi^2}{GM} +\end{aligned} +$$ + +--- + +Esto nos permite determinar la masa de un planeta conociendo el periodo y el radio orbital de uno de sus satélites: + +$$ +M = \frac{4\pi^2r^3}{GT^2} +$$ + +--- + +{{< figure library="true" src="interaccion-gravitatoria-2Bach-fisica/gravitacion-kepler.svg" width="70%" lightbox="false" >}} + +{{% /section %}} + +--- + +{{% section %}} + +## Momento angular + +Newton demostró que la fuerza gravitatoria es una [**fuerza central**](https://es.wikipedia.org/wiki/Fuerza_central), ya que está dirigida hacia un centro fijo (como el Sol) y depende únicamente de la distancia al mismo. + +--- + +Cuando las fuerzas son centrales, existe una magnitud, llamada [**momento angular**](https://es.wikipedia.org/wiki/Momento_angular), que se conserva, lo que facilita enormemente la descripción de muchos fenómenos asociados. + +--- + +Recordando que el momento lineal $\vec p = m\vec v$ representa la _inercia_ en un movimiento de traslación, se define el momento angular como la magnitud análoga para la rotación: + +$$ +\vec L = \vec r\times \vec p = \vec r\times m\vec v +$$ + +{{< figure library="true" src="interaccion-gravitatoria-2Bach-fisica/momento-angular.svg" width="60%" lightbox="false" >}} + +--- + +### Conservación del momento angular + +Una magnitud se conserva cuando permanece constante, es decir, no cambia con el tiempo. Estudiemos la variación de $\vec L$ con el tiempo para ver en qué circunstancias se conserva: + +{{< math width="100%" >}} +$$ +\frac{\mathrm d \vec L}{\mathrm d t} = \frac{\mathrm d}{\mathrm d t}\left(\vec r\times m\vec v\right) = \frac{\mathrm d \vec r}{\mathrm d t}\times m\vec v + \vec r\times m\frac{\mathrm d \vec v}{\mathrm d t} = \cancelto{0}{\vec v \times m \vec v} + \vec r\times m\vec a = \vec r\times \vec F +$$ +{{< /math >}} + +--- + +$\vec L$ se conserva si su variación con el tiempo es cero: + +{{< math width="100%" >}} +$$ +\frac{\mathrm d \vec L}{\mathrm d t} = \vec r\times \vec F = 0 \Leftrightarrow \begin{cases} +\vec F = 0 & \text{La fuerza neta que actúa sobre el cuerpo es cero.} \\ +\vec r\parallel \vec F & \text{Sucede en fuerzas centrales.} +\end{cases} +$$ +{{< /math >}} + +--- + +### Movimientos planetarios + +La conservación de $\vec L$ implica: + +- Dirección de $\vec L$ no cambia $\Rightarrow$ órbitas planas (plano $\perp$ a $\vec L$). +- Sentido de $\vec L$ no cambia $\Rightarrow$ planetas siempre giran en el mismo sentido. +- Módulo de $\vec L$ es constante: + + $$ + \begin{aligned} + L_1 &= L_2 \\ + r_1 \cancel mv_1\sin\theta_1 &= r_2 \cancel mv_2\sin\theta_2 + \end{aligned} + $$ + +--- + + En órbitas circulares: $r_1 = r_2; \theta_1 = \theta_2 = 90°\Rightarrow v_1 = v_2$ ($v$ constante). + + En órbitas elípticas: $r_\mathrm pv_\mathrm p = r_\mathrm av_\mathrm a$ (p $\rightarrow$ perihelio, a $\rightarrow$ afelio). + +--- + +### 2ª ley de Kepler + +La conservación de $\vec L$ también explica la [2ª ley de Kepler](#2ª-ley-ley-de-las-áreas). + +--- + +El área barrida en $\mathrm d t$ se puede aproximar por el área de un triángulo isósceles: + +{{< figure library="true" src="interaccion-gravitatoria-2Bach-fisica/kepler-L.svg" width="65%" lightbox="false" >}} + +--- + +\[ +\mathrm d A =\frac{1}{2}r\cdot r\sin\mathrm d \theta \ \simeq \frac{1}{2}r^2\mathrm d \theta +\] +\[ +\frac{\mathrm d A}{\mathrm d t} = \frac{1}{2}r^2\frac{\mathrm d \theta}{\mathrm d t} = \frac{1}{2}r^2\omega = \frac{1}{2}rv = \frac{L}{2m} +\] + +{{% /section %}} + +--- + +{{% section %}} + +## Campo gravitatorio + +- [Campo gravitatorio creado por una masa puntual](#/5/7) +- [Relación fuerza-campo](#/5/9) +- [Potencial gravitatorio](#/5/11) +- [Energía potencial gravitatoria](#/5/14) +- [Trabajo realizado por la fuerza gravitatoria](#/5/17) +- [Representación del campo gravitatorio](#/5/19) +- [Campo gravitatorio creado por distribuciones esféricas de masa](#/5/23) +- [Campo gravitatorio terrestre](#/5/30) + +(continúa hacia abajo) + +👇 + +--- + +La ley de gravitación universal introdujo la idea de [acción a distancia](https://es.wikipedia.org/wiki/Acción_a_distancia). ¿Cómo puede un objeto influenciar a otro sin estar en contacto con él? Hubo varios intentos de explicar cómo se producía la interacción a distancia. + +--- + +Algunos científicos afirmaban que existía una sustancia llamada [éter]() que llenaba el vacío (_fluido_) y era la responsable de transmitir la interacción. + +--- + +Este éter no se sabía de qué estaba hecho y su existencia no podía demostrarse experimentalmente, pero se mantuvo durante algún tiempo. + +--- + +Científicos posteriores demostraron que la interacción a distancia se producía también en el vacío, por lo que la existencia del éter no era necesaria. + +--- + +Finalmente, [Einstein](https://es.wikipedia.org/wiki/Albert_Einstein) justificó la acción a distancia por una perturbación en el [espacio-tiempo](https://es.wikipedia.org/wiki/Espacio-tiempo) debida a la presencia de una masa. Dicha perturbación se denominó [**campo gravitatorio**](https://es.wikipedia.org/wiki/Campo_gravitatorio). + +--- + +{{< figure library="true" src="interaccion-gravitatoria-2Bach-fisica/espacio-tiempo.jpg" >}} + +--- + +### Campo gravitatorio creado por una masa puntual + +El campo gravitatorio $\vec g$ creado por una [masa puntual](https://es.wikipedia.org/wiki/Masa_puntual) $M$ en un punto cualquiera que se encuentra a una distancia $r$ de ella viene dado por: + +$$ +\vec g = -\frac{GM}{r^2}\,\hat{\mathrm u}_\mathrm r +$$ + +y se mide en N/kg o m/s^2^ en el SI. + +--- + +#### Distribución de masas puntuales + +Cuando en una misma región del espacio existen varias ($n$) masas puntuales, el campo gravitatorio total es la suma vectorial de los campos creados por cada una de las masas, resultado que se conoce como [**principio de superposición**](https://es.wikipedia.org/wiki/Principio_de_superposición): +\[ +\vec g_\mathrm t = \sum_{i=1}^n \vec g_i = \sum_{i=1}^n \left(-\frac{GM_i}{r_i^2}\,\hat{\mathrm u}_{\mathrm r_i}\right) +\] + +--- + +### Relación fuerza-campo + +La fuerza $\vec F_\mathrm g$ que actúa sobre una masa $m$ colocada en un punto del espacio en el que existe un campo gravitatorio $\vec g$ viene dada por: +\[ +\vec F_\mathrm g = m\vec g, +\] + +--- + +expresión que coincide con la [**ley de gravitación universal de Newton**](#ley-de-gravitación-universal-de-newton): +\[ +\vec F_\mathrm g = m\vec g = -m\frac{GM}{r^2}\,\hat{\mathrm u}_\mathrm r = -\frac{GMm}{r^2}\,\hat{\mathrm u}_\mathrm r +\] + +--- + +### Potencial gravitatorio + +Como la **fuerza gravitatoria** es [**conservativa**](https://es.wikipedia.org/wiki/Fuerza_conservativa), el campo gravitatorio $\vec g$ puede expresarse en función de un [potencial gravitatorio](https://es.wikipedia.org/wiki/Potencial_gravitatorio) $V$. + +--- + +Este se define como el trabajo por unidad de masa que debe realizar una fuerza para traer un cuerpo desde el infinito (donde $V=0$ por convenio) hasta un punto P, a velocidad constante: +{{< math width="100%" >}} +\[ +V = \frac{W_{\infty\rightarrow P}}{m} = \frac{1}{m}\int_\infty^P \vec F\cdot \mathrm d \vec r = \frac{1}{m}\int_\infty^P \frac{GMm}{r^2}\,\mathrm d r = -\frac{GM}{r} +\] +{{< /math >}} + +tal que $\vec g = -\nabla V$. + +--- + +#### Distribución de masas puntuales +El potencial generado por una distribución de masas puntuales en un punto es la suma de los potenciales generados por cada una de las masas en dicho punto: +\[ +V_\mathrm t = \sum_{i=1}^n V_i = \sum_{i=1}^n \left(-\frac{GM_i}{r_i}\right) +\] + +--- + +### Energía potencial gravitatoria + +La energía potencial gravitatoria, $E_\mathrm p$, es la energía que adquiere una masa $m$ dentro de un campo gravitatorio $\vec g$. + +--- + +Se define como el trabajo que debe realizar una fuerza para traer un cuerpo desde el infinito (donde $V=0$ por convenio) hasta un punto P, a velocidad constante: +{{< math width="100%" >}} +\[ +E_\mathrm p = W_{\infty\rightarrow P} = \int_\infty^P \vec F\cdot \mathrm d \vec r = \int_\infty^P \frac{GMm}{r^2}\,\mathrm d r = -\frac{GMm}{r} +\] +{{< /math >}} + +--- + +También se puede calcular a partir del potencial gravitatorio: +\[ +E_\mathrm p = mV = -\frac{GMm}{r} +\] +El signo $-$ indica que la masa está ligada al campo gravitatorio (si $E_\mathrm p > 0$ estaría fuera del campo). + +--- + +### Trabajo realizado por la fuerza gravitatoria + +Cuando una masa $m$ se mueve libremente dentro de un campo gravitatorio, se realiza un trabajo $W_{1\rightarrow 2}$ tal que: +{{< math width="100%" >}} +\[ +\begin{aligned} +W_{1\rightarrow 2} = \int_1^2 \vec F\cdot \mathrm d \vec r = \int_1^2 -F\,\mathrm d r &= \int_1^2 -\frac{GMm}{r^2}\,\mathrm d r \\ +&= \left. GMm\frac{1}{r}\right |_1^2 = \frac{GMm}{r_2} - \frac{GMm}{r_1} =-\symup\Delta E_\mathrm p = -m\symup\Delta V, +\end{aligned} +\] +{{< /math >}} + +que solo depende de las posiciones inicial y final, lo que significa que $\vec F_\mathrm g$ es **conservativa**. + +--- + +{{< math width="100%" >}} +\[ +\begin{cases} +W=0 & \text{La masa $m$ describe una trayectoria cerrada (o se mueve por una superficie equipotencial).} \\ +W>0 & \text{La masa $m$ se acerca a $M$, desplazándose hacia $V$ decrecientes (proceso espontáneo).} \\ +W<0 & \text{La masa $m$ se aleja de $M$ (proceso no espontáneo, se requiere una $\vec F_\mathrm{externa}$).} +\end{cases} +\] +{{< /math >}} + +--- + +### Representación del campo gravitatorio + +Existen dos maneras de representar el campo gravitatorio: + +- [Líneas de campo](#/5/20) +- [Superficies equipotenciales](#/5/21) + +(continúa hacia abajo) + +👇 + +--- + +#### Líneas de campo + +Son tangentes al campo gravitatorio en cada punto. Cumplen que: + +- Nunca se cortan (supondría que en un mismo punto habría dos valores de campo, lo que es imposible). +- El número de líneas es proporcional a la intensidad del campo gravitatorio. + +--- + +#### Superficies equipotenciales + +Son regiones en las que el potencial gravitatorio tiene el mismo valor. Cumplen que: + +- Nunca se cortan. +- Siempre son perpendiculares a las líneas de campo. + +--- + +{{< figure library="true" src="interaccion-gravitatoria-2Bach-fisica/campo-tierra-luna.svg" width="100%" lightbox="false" >}} + +--- + +### Campo gravitatorio creado por distribuciones esféricas de masa + +Hasta ahora hemos visto el campo creado por masas puntuales, sin extensión. Podemos aproximar que una masa es puntual cuando sus dimensiones son mucho menores que la distancia considerada. Es decir, cuando trabajamos en regiones del campo que están alejadas de la masa que lo genera; por ejemplo, en el Sistema Solar. + +--- + +Cuando las dimensiones no son despreciables frente a la distancia, hay que considerar el tamaño y la geometría del objeto. Esto sucede, por ejemplo, cuando consideramos el campo en la superficie de terrestre o en sus proximidades. + +--- + +En ese caso, para el cálculo de $\vec g$ se requiere emplear un teorema matemático: el [**teorema de Gauss**](https://es.wikipedia.org/wiki/Ley_de_Gauss). + +--- + +Antes de enunciar el teorema hay que definir el [**flujo**](https://es.wikipedia.org/wiki/Flujo) $\symup\Phi = \oint_S \vec g \cdot \mathrm d \vec s$, como el número de líneas de campo que atraviesa una superficie. + +{{< figure library="true" src="interaccion-gravitatoria-2Bach-fisica/flujo.svg" width="75%" lightbox="false" >}} + +--- + +El [**teorema de Gauss**](https://es.wikipedia.org/wiki/Ley_de_Gauss) nos permite relacionar el flujo neto $\Phi$ que atraviesa una superficie cerrada $S$ con la masa encerrada $M_\mathrm{encerrada}$ por dicha superficie: +\[ +\Phi = -4\pi GM_\mathrm{encerrada} +\] + +--- + +#### Campo creado por una esfera hueca + +Una esfera que solo tiene masa en su superficie: + +{{< figure library="true" src="interaccion-gravitatoria-2Bach-fisica/esfera-hueca.svg" width="100%" lightbox="false" theme="light" >}} + +--- + +#### Campo creado por una esfera maciza homogénea + +Una esfera con densidad homogénea $\rho$: + +- Para $r \leq R$: + {{< math width="100%" >}} + \left.\begin{aligned} + \Phi &= \oint_S \vec g \cdot \mathrm d \vec s = -g\oint_S \mathrm d s = -g\cdot 4\pi r^2 \\ + \Phi &= -4\pi G\cdot M_\mathrm{encerrada} = -4\pi G\rho\cdot\frac{4}{3}\pi r^3 + \end{aligned}\right\rbrace \vec g = -\dfrac{4\pi G\rho}{3}r\,\hat{\mathrm u}_\mathrm r\ (\text{proporcional al radio $r$}) + {{< /math >}} + +- Para $r \geq R$: + + {{< math width="100%" >}} + \left.\begin{aligned} + \Phi &= \oint_S \vec g \cdot \mathrm d \vec s = -g\oint_S \mathrm d s = -g\cdot 4\pi r^2 \\ + \Phi &= -4\pi G\cdot M + \end{aligned}\right\rbrace \vec g = -\frac{GM}{r^2}\,\hat{\mathrm u}_\mathrm r\ (\text{igual que masa puntual}) + {{< /math >}} + +--- + +### Campo gravitatorio terrestre + +Lo que acabamos de ver para una esfera maciza homogénea también es aplicable a esferas con capas homogéneas concéntricas, por lo que podemos aplicarlo a la Tierra: +$$ +\vec g = -\frac{GM_\mathrm T}{r^2}\,\hat{\mathrm u}_\mathrm r \quad \mathrm{para}\ r=R_\mathrm T+h\geq R_\mathrm T\ (h\geq 0) +$$ + +--- + +La realidad es un poco más complicada. A día de hoy tenemos el llamado [Modelo de Referencia Preliminar](https://es.wikipedia.org/wiki/Modelo_de_referencia_preliminar_de_la_Tierra) (PREM por sus siglás en inglés): +{{< figure library="true" src="interaccion-gravitatoria-2Bach-fisica/g-tierra.svg" width="80%" lightbox="false" >}} + +--- + +En esta **práctica virtual** diseñada para 1.º de Bachillerato, el [Departamento de Física y Química del IES Valle del Saja](http://www.fqsaja.com) nos propone dos métodos bien diferentes pero igual de eficaces para **determinar** el valor de la **aceleración** de la **gravedad terrestre**: + +--- + +{{< youtube WZRkV4pMB1Q >}} + +[Aquí](https://drive.google.com/file/d/1deVQJYVJxXDQzQDwnWrakmMgcg5WQ4mY/view) puedes descargarte el **guion** de la **práctica**. + +--- + +#### Variación de $\vec g$ con la altura + +En la superficie terrestre $r = R_\mathrm T$ y por lo tanto: +$$ +g_0 = \frac{GM_\mathrm T}{R_\mathrm T^2} +$$ + +A una altura $h$ sobre la superficie: +{{< math width="100%" >}} +$$ +g = \frac{GM_\mathrm T}{\left(R_\mathrm T+h\right)^2} = g_0\frac{R_\mathrm T^2}{\left(R_\mathrm T+h\right)^2} = g_0\frac{1}{\left(1+\dfrac{h}{R_\mathrm T}\right)^2} +$$ +{{< /math >}} + +--- + +#### Variación de $\vec g$ con la latitud + +Al igual que con la altura, la gravedad varía con la latitud. Consideremos un punto P tal como indica la figura: + +{{< figure library="true" src="interaccion-gravitatoria-2Bach-fisica/g-latitud.svg" width="55%" lightbox="false" theme="light" >}} + +--- + +El punto P gira debido a la rotación de la Tierra, describiendo una circunferencia de radio $r$. + +--- + +Al ser un [sistema de referencia no inercial](https://es.wikipedia.org/wiki/Sistema_de_referencia_no_inercial), aparece una aceleración [centrífuga](https://es.wikipedia.org/wiki/Fuerza_centrífuga) ([ficticia](https://es.wikipedia.org/wiki/Fuerza_ficticia)), la cual resulta conveniente descomponer en ejes radial y tangencial: +$$ +\vec a_\mathrm{cf} = a_{\mathrm{cf}_\mathrm r}\,\hat{\mathrm u}_\mathrm r + a_{\mathrm{cf}_\mathrm t}\,\hat{\mathrm u}_\mathrm t, +$$ +donde $a_{\mathrm{cf}_\mathrm r} = a_\mathrm{cf}\cos\varphi = \omega^2 r \cos\varphi = \omega^2R_\mathrm T\cos^2\varphi$. + +--- + +El punto P siente una gravedad efectiva inferior ya que la componente radial algo contrarresta: +{{< math width="100%" >}} +$$ +g_\mathrm{efectiva} = g-a_{\mathrm{cf}_\mathrm r} = g-\omega^2R_\mathrm T\cos^2\varphi = \begin{cases} + g-\omega^2 R_\mathrm T & \varphi=0°\ (\text{ecuador}) \\ + g & \varphi=90°\ (\text{polos}) + \end{cases} +$$ +{{< /math >}} + +--- + +{{< figure library="true" src="interaccion-gravitatoria-2Bach-fisica/variacion-g-altura-latitud.svg" width="90%" lightbox="false" theme="light" >}} + +--- + +#### ¿Cómo se clasifican las órbitas de los satélites que giran en torno a la Tierra? + +- [En función de la latitud](#/5/41) +- [En función de su forma](#/5/42) +- [En función de la altura sobre la superficie](#/5/43) + +(continúa hacia abajo) + +👇 + +--- + +##### En función de la latitud +- Órbitas polares ($\varphi = 90°$). +- Órbitas ecuatoriales ($\varphi = 0°$). +- Órbitas inclinadas. + +--- + +##### En función de su forma +- Circulares. +- Elípticas (cobertura a mayor altitud). +- No geocéntricas (exploraciones interplanetarias). + +--- + +##### En función de la altura sobre la superficie + +{{< svg "static/media/interaccion-gravitatoria-2Bach-fisica/tipos-orbitas.svg" >}} + +{{% /section %}} + +--- + +{{% section %}} + +## Movimiento de los satélites en órbitas + +- [Satélites geoestacionarios](#/6/3) +- [Energía de los satélites](#/6/5) +- [Energía de los satélites y tipo de órbita](#/6/12) + +(continúa hacia abajo) + +👇 + +--- + +Supongamos que un satélite describe una órbita circular a una altura $h$ ($r=R+h$) sobre la superficie de un planeta, con [MCU]({{< relref "apuntes-cuarto-eso/movimientos/index.md#movimiento-circular-uniforme-mcu" >}}). Si el satélite permanece en la órbita, debe cumplirse: +$$ +\begin{aligned} +F_\mathrm g & = F_\mathrm c \\ +\frac{GM\cancel m}{r^{\cancel 2}} & = \frac{\cancel mv^2}{\cancel r} +\end{aligned} +$$ + +--- + +de donde podemos obtener: + +- Velocidad orbital: $\displaystyle v = \sqrt{\frac{GM}{R+h}}$ +- Periodo de revolución: $\displaystyle T = 2\pi\sqrt{\frac{\left(R+h\right)^3}{GM}}$ +- Frecuencia: $f=1/T$ + +--- + +### Satélites geoestacionarios +Son aquellos cuyo periodo coincide con el [periodo de rotación terrestre](https://es.wikipedia.org/wiki/Día), de forma que orbitan manteniéndose siempre en la misma vertical (aparentemente no se mueven). Se colocan en el plano del ecuador para que $\vec g$ no varíe con la latitud. + +--- + +Despejando de la [3ª ley de Kepler](#3ª-ley-ley-de-los-periodos): +{{< math width="100%" >}} +$$ +\frac{T^2}{r^3} = \frac{4\pi^2}{GM_\mathrm T} \rightarrow r = \sqrt[3]{\frac{GM_\mathrm T}{4\pi^2}} \rightarrow h = r-\mathrm R_T \approx 35\,800\,\mathrm{km} +$$ +{{< /math >}} + +--- + +### Energía de los satélites + +Cuando describe una órbita, la energía mecánica de un satélite viene dada por: +{{< math width="100%" >}} +\[ +E = E_\mathrm c + E_\mathrm p = \frac{1}{2}mv^2 - \frac{GMm}{r} = \frac{1}{2}\frac{GMm}{r} - \frac{GMm}{r} = -\frac{1}{2}\frac{GMm}{r} = \frac{E_\mathrm p}{2} +\] +{{< /math >}} + +--- + +Si tenemos en cuenta que todas las **fuerzas** que actúan son **conservativas**, la **energía mecánica se conserva**. + +--- + +Así, es posible calcular: + +Velocidad de lanzamiento +: Es la velocidad que hay que comunicar a un satélite para colocarlo en su órbita, suponiendo que se lanza desde la superficie terrestre: +{{< math width="100%" >}} +$$ +\begin{aligned} +E_1 & = E_2 \\ +E_{\mathrm c_1} + E_{\mathrm p_1} & = \frac{E_{\mathrm p_2}}{2} \\ +\frac{1}{2}\cancel mv^2 - \frac{GM_\mathrm T \cancel m}{R_\mathrm T} & = -\frac{GM_\mathrm T \cancel m}{2r} \rightarrow v = \sqrt{2GM_\mathrm T\left(\frac{1}{R_\mathrm T}-\frac{1}{2r}\right)} +\end{aligned} +$$ +{{< /math >}} + +--- + +Velocidad de escape +: Si en el lanzamiento proporcionamos al satélite una velocidad suficientemente elevada, este puede llegar a escapar del campo gravitatorio terrestre. A esta (mínima) velocidad se le denomina [**velocidad de escape**](https://es.wikipedia.org/wiki/Velocidad_de_escape). + +--- + +Imponiendo la condición de que el objeto se aleje hasta una distancia infinita y quede en reposo ($E=0$): +$$ +\begin{aligned} +E & = 0 \\ +\frac{1}{2}mv_\mathrm e^2 - \frac{GM_\mathrm T m}{R_\mathrm T} & = 0 \rightarrow v_\mathrm e = \sqrt{\frac{2GM_\mathrm T}{R_\mathrm T}} +\end{aligned} +$$ + +Para $v\geq v_\mathrm e$ el satélite escapa de la influencia de $\vec g$ y se mueve libremente. Es importante destacar, a la vista de la expresión, que solo depende de $M_\mathrm T$ y de $R_\mathrm T$ (es independiente de las características del satélite). + +--- + +Trabajo para pasar de una órbita a otra +: Supongamos que queremos pasar un satélite de una órbita 1 a otra órbita 2. Será necesario realizar un trabajo tal que: +{{< math width="100%" >}} +\[ +E_1+W = E_2 \rightarrow W = E_2-E_1 = - \frac{GM_\mathrm T m}{2r_2}-\left(- \frac{GM_\mathrm T m}{2r_1}\right) = \frac{GM_\mathrm T m}{2}\left(\frac{1}{r_1}-\frac{1}{r_2}\right) +\] +{{< /math >}} + +--- + +{{< figure library="true" src="interaccion-gravitatoria-2Bach-fisica/orbita-hohmann.svg" width="70%" lightbox="false" theme="light" >}} + +--- + +### Energía de los satélites y tipo de órbita +La energía de un satélite determina el tipo de órbita que describe: + +\[ +E = \overbrace{\frac{1}{2}mv^2}^{+}\overbrace{-\frac{GMm}{r}}^{-} +\] + +--- + +$E<0$ Ligado al campo (órbitas cerradas): +{{< math width="100%" >}} +$$ +\begin{cases} +E_\mathrm c = 0 & \text{El satélite cae por atracción gravitatoria.} \\ +E_\mathrm c \neq 0 & \text{Describe una órbita cerrada (circular o elíptica). Ej. satélites o planetas.} +\end{cases} +$$ +{{< /math >}} + +--- + +$E\geq 0$ No ligado al campo (órbitas abiertas): +{{< math width="100%" >}} +\[ +\begin{cases} +E = 0 & \text{Órbita parabólica: $E_\mathrm c = -E_\mathrm p$. Ej. cometas.} \\ +E > 0 & \text{Órbita hiperbólica: $E_\mathrm c > E_\mathrm p$. Ej. asteroides.} +\end{cases} +\] +{{< /math >}} + +--- + + + +--- + + + +{{% /section %}} +
## Exportar a PDF diff --git a/layouts/shortcodes/math.html b/layouts/shortcodes/math.html new file mode 100644 index 000000000..194c13fcf --- /dev/null +++ b/layouts/shortcodes/math.html @@ -0,0 +1,127 @@ +{{- /* Last modified: 2024-05-07T07:11:34-07:00 */}} + +{{- /* +Copyright 2023 Veriphor LLC + +Licensed under the Apache License, Version 2.0 (the "License"); you may not +use this file except in compliance with the License. You may obtain a copy of +the License at + +https://www.apache.org/licenses/LICENSE-2.0 + +Unless required by applicable law or agreed to in writing, software +distributed under the License is distributed on an "AS IS" BASIS, WITHOUT +WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the +License for the specific language governing permissions and limitations under +the License. +*/}} + +{{- /* +Renders an SVG image of a mathematical expression or equation from LaTeX using the Math API service. + +References: + + - https://math.vercel.app/ + - https://github.com/uetchy/math-api/ + - https://docs.mathjax.org/ + +@context {string} Inner The content between the opening and closing shortcode tags. +@context {string} InnerDeindent The content between the opening and closing shortcode tags with indentation removed. +@context {string} Name The file name of the shortcode template, excluding the extension. +@context {int} Ordinal The zero-based ordinal of the shortcode on the page, or within its parent shortcode. +@context {page} Page A reference to the page containing the shortcode. +@context {map} Params The parameters specified in the opening shortcode tag. +@context {hugolib.ShortcodeWithPage} Parent The context of the parent shortcode. +@context {text.Position} Position The position of the shortcode within the page content. + +@method {any} Get Returns the parameter value for the given key (for named parameters) or position (for positional parameters). +@mathod {bool} IsNamedParams Returns true if the shortcode is called with named instead of positional parameters. +@method {maps.Scratch) Scratch Returns a writable Scratch to store and manipulate data. + +@param {string} [Params.class] A class name to add to the class attribute of the wrapping span element. +@param {string} [Params.color=black] The foreground color specified as a named CSS color or hex value. +@param {string} [Params.id] The id attribute of the wrapping span element. + +@returns {template.html} + +@example {{< math >}}${(x+y)}^2${{< /math >}} +*/}} + +{{- /* Verify minimum required version. */}} +{{- $minHugoVersion := "0.125.6" }} +{{- if lt hugo.Version $minHugoVersion }} + {{- errorf "The %s shortcode requires Hugo v%s or later." .Name $minHugoVersion }} +{{- end }} + +{{- /* Get context. */}} +{{- $inner := trim .Inner "\n\r" }} +{{- $name := .Name }} +{{- $ordinal := .Ordinal }} +{{- $position := .Position }} + +{{- /* Initialize. */}} +{{- $apiEndpoint := "https://math.vercel.app/" }} +{{- $color := .Get "color" | default "" }} + +{{- /* Determine display mode. */}} +{{- $displayMode := true }} +{{- if or (strings.HasPrefix $inner `$$`) (strings.HasPrefix $inner `\[`) }} + {{- $displayMode = true }} +{{- else if or (strings.HasPrefix $inner `$`) (strings.HasPrefix $inner `\(`) }} + {{- $displayMode = false }} +{{- end }} + +{{- /* Strip display mode indicators. */}} +{{- $inner = trim $inner `$` }} +{{- $inner = $inner | strings.TrimPrefix `\[` }} +{{- $inner = $inner | strings.TrimPrefix `\(` }} +{{- $inner = $inner | strings.TrimSuffix `\]` }} +{{- $inner = $inner | strings.TrimSuffix `\)` }} + +{{- /* Determine class attribute. */}} +{{- $class := "math math-inline" }} +{{- if $displayMode }} + {{- $class = "math math-block" }} +{{- end }} +{{- with .Get "class" }} + {{- $class = printf "%s %s" $class . }} +{{- end }} + +{{- /* Get id attribute. */}} +{{- $id := or (.Get "id") (printf "h-sc-%s-%d" $name $ordinal) }} + +{{- /* Define attributes map. */}} +{{- $attrs := dict "class" $class "id" $id }} + +{{- /* Create query string. */}} +{{- $mode := "inline" }} +{{- if $displayMode }} + {{- $mode = "from" }} +{{- end }} +{{- $qs := querify $mode $inner "color" $color }} + +{{- /* Get image. */}} +{{- $url := printf "%s?%s" $apiEndpoint $qs }} +{{- with resources.GetRemote $url }} + {{- with .Err }} + {{- errorf "The %q shortcode was unable to get the remote image. See %s. %s" $name $position . }} + {{- else }} + {{- $url = .RelPermalink }} + {{- end }} +{{- else }} + {{- errorf "The %q shortcode was unable to get the remote image. See %s" $name $position }} +{{- end }} + +{{- /* Render. */}} + + mathematical expression or equation + +{{- /**/ -}} \ No newline at end of file diff --git a/static/media/interaccion-gravitatoria-2Bach-fisica/tipos-orbitas.svg b/static/media/interaccion-gravitatoria-2Bach-fisica/tipos-orbitas.svg index 22e93c087..d9c5fc70f 100644 --- a/static/media/interaccion-gravitatoria-2Bach-fisica/tipos-orbitas.svg +++ b/static/media/interaccion-gravitatoria-2Bach-fisica/tipos-orbitas.svg @@ -1,4 +1,4 @@ - + Comparación entre entre las órbitas de varios satélites Comparación de las órbitas geoestacionaria, GPS, GLONASS, Galileo y COMPASS (satélites de órbita media), Estación Espacial Internacional, Telescopio Espacial Hubble, Constelación Iridium y órbita cementerio con los cinturones de radiación de Van Allen y el tamaño nominal de la tierra, ilustración de CMG Lee. Bitmap de la Tierra http://commons.wikimedia.org/wiki/File:North_pole_february_ice-pack_1978-2002.png by Geo Swan. La órbita de la Luna es cerca de 9 veces más grande (en radio y largo) que la órbita geoestacionaria. Esta animación se mueve a 1 hora por segundo. En el archivo SVG, se puede posicionar el mouse sobre una órbita o su nombre para resaltarla y hacer clic para ir a su respectivo articulo en la Wikipedia.