Grupo I secc: 09 MATRICULAS : BB‐8755, AD‐6247, BF‐9375, CB‐2308, CA‐8792, CB‐0233, CI‐6080
MECÁNI CA DEL SI SI STE STEMA MA DE PARTÍ CULAS Empecemos recordando la mecánica de una partícula, para después extenderla a un sistema de muchas partículas. Recordamos la Cinemática (descripción del movimiento). Primero posicionamos la partícula (que suponemos puntual) en un Sistema de Referencia:
Llamaremos al vector de posición: . A partir de él definiremos los vectores velocidad y aceleración que miden la variación temporal de la posición y la velocidad, respectivamente:
(Repasad el álgebra vectorial).
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Grupo I secc: 09 MATRICULAS : BB‐8755, AD‐6247, BF‐9375, CB‐2308, CA‐8792, CB‐0233, CI‐6080
A partir de la aceleración se pueden calcular velocidades y trayectorias (posiciones en función del tiempo), aunque hay que conocer velocidad y posición en algún instante de tiempo (en cierto
serán datos).
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Grupo I secc: 09 MATRICULAS : BB‐8755, AD‐6247, BF‐9375, CB‐2308, CA‐8792, CB‐0233, CI‐6080
Pero la mecánica no se conforma con describir el movimiento, se busca su relación con las causas que lo producen: las Fuerzas. Toda la mecánica se desarrolla a partir de las: LEYES DE NEWTON 1) LEY DE LA INERCIA: Si sobre una partícula no actúan fuerzas, si estaba en reposo continúa en reposo y si estaba moviéndose con una velocidad v r , continúa haciéndolo con la misma velocidad. 2) Si sobre una partícula actúan fuerzas, siendo F su resultante ( F experimenta una aceleración dada por:
∑Fi
r r ), la partícula
3) LEY DE ACCIÓN Y REACCIÓN: Si una partícula 1 hace una fuerza 21 F r sobre la partícula 2, ésta hace sobre la primera una fuerza 12 F r que tiene el mismo módulo, la misma dirección y sentido opuesto:
Hemos introducido una magnitud fundamental para la dinámica, la MASA, que deberíamos llamar masa inerte, ya que mide la oposición de los cuerpos a cambios en su estado de reposo o movimiento y que no hay que confundir con el peso. Para la dinámica resulta interesante introducir una nueva magnitud, EL MOMENTO LINEAL O CANTIDAD DE MOVIMIENTO, que nos permite expresar la segunda ley de Newton de una forma equivalente:
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Grupo I secc: 09 MATRICULAS : BB‐8755, AD‐6247, BF‐9375, CB‐2308, CA‐8792, CB‐0233, CI‐6080
Análogamente la primera ley de Newton nos dirá que si sobre una partícula no actúan fuerzas, su momento lineal es una constante del movimiento, es decir se conserva. Éste es el teorema de conservación del momento lineal para una partícula:
SISTEMA DE MUCHAS PA RTÍCULAS Para describir un sistema de N partículas necesitamos N Ecuaciones de Newton:
Para cada partícula i (i=1,…,N) tenemos que sumar las fuerzas que realizan sobre ella las otras N-1 partículas y además las fuerzas exteriores que actúan sobre ella. Veremos que su dinámica resulta simplificada si estudiamos el sistema de una forma global. Vamos a plantearlo para un sistema de dos partículas, pero resulta válido para sistemas de N partículas en general. Definimos el momento lineal total del sistema:
En ausencia de fuerzas exteriores, se conserva el momento lineal total, ya que las fuerzas interiores cumplen el principio de acción y reacción.
TEOREMA DE CONSE CONSERVACIÓN RVACIÓN DEL MOMENTO LI NEAL AP LI CADO A UN SISTEMA SISTEMA DE PARTÍCULAS.
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Grupo I secc: 09 MATRICULAS : BB‐8755, AD‐6247, BF‐9375, CB‐2308, CA‐8792, CB‐0233, CI‐6080
Asimismo se generaliza la segunda ley de Newton a un sistema de partículas:
Y esta descripción del sistema de partículas como un todo lleva a la definición del CENTRO DE MASAS del sistema: un punto imaginario, cuyo movimiento se describe como si tuviera ∑mi) y estuviera sometido únicamente a concen concentra trada da en él él toda toda la masa masa del sist sistema ema (M (M la acción de las fuerzas exteriores. Este Centro de masas se calcula:
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Grupo I secc: 09 MATRICULAS : BB‐8755, AD‐6247, BF‐9375, CB‐2308, CA‐8792, CB‐0233, CI‐6080
De este modo se puede describir el movimiento global del sistema como si se tratara de una única partícula, caracterizada caracterizada por el momento total del sistema, sometida a la acción de las No es una fuerzas exteriores, de masa M y cuya posición la describe descripción completa del movimiento, ya que las partículas del sistema se moverán respecto al CDM (a causa de las fuerzas interiores). (Ejemplo del saltador de trampolín: el CDM describe una parábola ya que sobre él actúa la fuerza de la l a gravedad, las partes del cuerpo giran respecto al CDM de forma complicada a causa de las fuerzas interiores).
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Grupo I secc: 09 MATRICULAS : BB‐8755, AD‐6247, BF‐9375, CB‐2308, CA‐8792, CB‐0233, CI‐6080
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Grupo I secc: 09 MATRICULAS : BB‐8755, AD‐6247, BF‐9375, CB‐2308, CA‐8792, CB‐0233, CI‐6080
mediante las TRANSFORMACIONES TRANSFORMACIONES DE GALILEO entre sistemas de referencia inerciales (i.e. que se mueven con velocidad relativa constante entre ellos).
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Grupo I secc: 09 MATRICULAS : BB‐8755, AD‐6247, BF‐9375, CB‐2308, CA‐8792, CB‐0233, CI‐6080
Comprobad que si se calcula el momento lineal total de un sistema de partículas en el sistema centro de masas resulta: P*=0. CONSIDERACIONES ENERGÉTICAS
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Grupo I secc: 09 MATRICULAS : BB‐8755, AD‐6247, BF‐9375, CB‐2308, CA‐8792, CB‐0233, CI‐6080
el de las FUERZAS CONSERVATIVAS, que cumplen que el trabajo no depende del camino seguido, solamente depende de los puntos inicial y final de la trayectoria. Otra forma equivalente de definirlas, es que el trabajo realizado por la fuerza para, saliendo de un punto determinado, volver al mismo punto es idénticamente nulo.
De una forma general, se puede demostrar que:
Recordad que en términos de las coordenadas intrínsecas (normal y tangencial): la fuerza se expresa como:
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Grupo I secc: 09 MATRICULAS : BB‐8755, AD‐6247, BF‐9375, CB‐2308, CA‐8792, CB‐0233, CI‐6080
Se cumplirá que el trabajo realizado por una fuerza es igual a la variación de energía cinética. Volved a pensar en el significado del signo del trabajo. Cuando pasamos a un sistema de partículas, todas estas nuevas magnitudes se extienden de una forma natural. Así por ejemplo, la energía cinética de un sistema de partículas no es más que la suma de las l as energías cinéticas de todas las partículas del sistema.
Suele ser conveniente expresarla en función de la energía cinética del Centro de Masas y la relativa al CDM, que resulta de sustituir la velocidad en el sistema de referencia de partida por la velocidad en el sistema centro de masas más la velocidad relativa entre los dos sistemas (ver la transformación de velocidades entre sistemas de referencia inerciales):
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Grupo I secc: 09 MATRICULAS : BB‐8755, AD‐6247, BF‐9375, CB‐2308, CA‐8792, CB‐0233, CI‐6080
Eligiendo el camino como nos interese dado que al ser la fuerza conservativa el resultado no depende del mismo. Esta definición de la energía potencial nos permite:
Por lo que si definimos una nueva magnitud, LA ENERGÍA MECÁNICA, Em=Ec+U, habremos llegado al TEOREMA DE CONSERVACIÓN CONSERVACIÓN DE LA ENERGÍA MECÁNICA: la energía mecánica se conserva si todas las fuerzas que actúan sobre la partícula/sistema son conservativas. conservativas. En dicho caso las energías cinéticas y potenciales son intercambiables y la energía puede expresarse en forma de una u otra a lo largo del movimiento. No hay que olvidar que la energía total se conserva siempre (la energía ni se crea ni se destruye, solamente se transforma). La energía mecánica no coincide con la energía total, hay otras formas de energía: química, luminosa, sonora, térmica, etc. En un caso general, con fuerzas conservativas y no conservativas, el trabajo seguiría siendo igual a la variación en la energía cinética que produce, sin embargo, sólo las fuerzas conservativas tendrían asociada energía potencial y cumplirían que:
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Grupo I secc: 09 MATRICULAS : BB‐8755, AD‐6247, BF‐9375, CB‐2308, CA‐8792, CB‐0233, CI‐6080
y tiene un significado claro cuando se representa gráficamente la curva de energía potencial. La fuerza da la pendiente de la recta tangente a la curva de energía potencial en cada punto cambiada de signo. Así, si buscamos las POSICIONES DE EQUILIBRIO estaremos buscando cuándo se hace cero la fuerza y por lo tanto cuándo es horizontal la tangente a la curva, o lo que es lo mismo, cuando vale cero su primera derivada. Esta condición la cumplen máximos, mínimos y puntos de inflexión de la energía potencial que corresponden a situaciones de equilibrio inestable, estable y neutro.
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Grupo I secc: 09 MATRICULAS : BB‐8755, AD‐6247, BF‐9375, CB‐2308, CA‐8792, CB‐0233, CI‐6080
Vemos que resulta sencillo expresarlo en términos de la velocidad angular (basta con pasar las coordenadas cartesianas a coordenadas polares) y una nueva magnitud que desempeña un papel equivalente a la masa en la traslación: el MOMENTO DE INERCIA, I.
Mide la oposición a los cambios en el estado de rotación, y aunque depende de la masa, lógicamente, resulta importante la distribución de la misma respecto al eje de giro elegido. Es, pues, una propiedad de cada sistema de partículas, pero depende del eje de rotación que se elija.
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Grupo I secc: 09 MATRICULAS : BB‐8755, AD‐6247, BF‐9375, CB‐2308, CA‐8792, CB‐0233, CI‐6080
Y se generalizan análogamente los teoremas de conservación. Si las fuerzas no realizan momento, se conserva el momento angular total del sistema de partículas, lo que a su vez implic implicaa que que si I aume aumenta nta,, dismin disminuye uye y vicev vicevers ersaa (pens (pensad ad en en el ejem ejemplo plo de de las las piruet piruetas as de una patinadora sobre hielo; se puede considerar que el patín desliza sobre el hielo sin rozar apenas). Si nos replanteamos en el sistema de partículas la diferencia que hay entre las fuerzas interiores y las exteriores en lo que a la rotación se refiere, podemos comprobar que las
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Grupo I secc: 09 MATRICULAS : BB‐8755, AD‐6247, BF‐9375, CB‐2308, CA‐8792, CB‐0233, CI‐6080
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Grupo I secc: 09 MATRICULAS : BB‐8755, AD‐6247, BF‐9375, CB‐2308, CA‐8792, CB‐0233, CI‐6080
Si hacemos los mismos cálculos para la situación en que dos masas, situadas a distancia constante entre ellas, giran respecto al eje Z (ver figura), observaremos observaremos que en este caso el momento angular calculado respecto a O es el mismo que el calculado respecto a O’, y resulta paralelo al eje Z y por lo tanto a la velocidad angular. Comprobad este resultado.
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Grupo I secc: 09 MATRICULAS : BB‐8755, AD‐6247, BF‐9375, CB‐2308, CA‐8792, CB‐0233, CI‐6080
Comprobad dicho resultado. Un ejemplo claro para verlo es en el de la rotación de la l a Tierra. La Tierra rota alrededor del Sol y simultáneamente, en torno a su eje, de tal forma que el momento angular de la Tierra proviene de ambas rotaciones. Resulta usualmente mucho más sencillo describir el movimiento de la Tierra como resultado de superponer por un lado la rotación en torno al Sol, que se puede describir simplemente como rotación del CDM que, tendrá asociado un ón
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Grupo I secc: 09 MATRICULAS : BB‐8755, AD‐6247, BF‐9375, CB‐2308, CA‐8792, CB‐0233, CI‐6080
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Grupo I secc: 09 MATRICULAS : BB‐8755, AD‐6247, BF‐9375, CB‐2308, CA‐8792, CB‐0233, CI‐6080
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Grupo I secc: 09 MATRICULAS : BB‐8755, AD‐6247, BF‐9375, CB‐2308, CA‐8792, CB‐0233, CI‐6080
Un tipo especial de movimiento en el que podemos aplicar lo visto hasta ahora de la dinámica es en el producido por FUERZAS CENTRALES: CENTRALES: aquellas cuya dirección siempre apunta hacia un mismo punto, el centro de la fuerza. Con respecto a dicho punto las fuerzas centrales nunca realizan momento:
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Grupo I secc: 09 MATRICULAS : BB‐8755, AD‐6247, BF‐9375, CB‐2308, CA‐8792, CB‐0233, CI‐6080
molécula tendrá en rotación una distancia interatómica diferente a la correspondiente a ausencia de rotación (rmin), en concreto la distancia interatómica corresponderá a r > rmin dado que la fuerza debe ser atractiva (tratad de entender el sentido de la fuerza que corresponde a valores de r menores y mayores de rmin). Para obtener la nueva posición de equilibrio basta con minimizar el POTENCIAL EFECTIVO:
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Grupo I secc: 09 MATRICULAS : BB‐8755, AD‐6247, BF‐9375, CB‐2308, CA‐8792, CB‐0233, CI‐6080
Recordad que para una molécula diatómica m sería la masa reducida del sistema. Otro ámbito de aplicación de todo lo que hemos visto es el de las colisiones. COLISIONES
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( A ) SISTEMA FUERZA EXTERNAS ( B ) SISTEMA FUERZA EFECTIVAS ( C) RESULTANTE FUERZA EXTERNAS O SISTEMA FUERZA PAR EQQ EN EL C.D.M ( D) RESULTANTE SISTEMA FUERZAS EFECTIVAS
La relación de equivalencia entre los sistemas de fuerzas extremas y efectivas se llama Principio de D’ alembert. El mismo puede asumirse como el DCL de ecuaciones básicas ( A ) Y ( B ). El principio de d'Alembert, d'Alembert, enunciado por Jean d'Alembert en su obra maestra Tratado de dinámica de 1743, establece que la suma de las fuerzas externas que actúan sobre un cuerpo y las denominadas fuerzas de inercia forman un sistema de fuerzas en equilibrio. A este equilibrio se le denomina equilibrio dinámico. dinámico. El principio de d'Alembert establece que para todas las fuerzas de un sistema:
Donde la suma se extiende sobre todas las partículas del sistema, siendo: , momento de la partícula i-ésima. , fuerza sobre la partícula i-ésima. cualquier campo vectorial de desplazamientos virtuales sobre el conjunto de partículas que sea compatible con los enlaces y restricciones restricciones de movimiento existentes. El principio de d'Alembert es realmente una generalización de la segunda ley de
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D'Alembert durante el resto de su vida y de manera especial en su Mécanique Analytique. Analytique. Tal cambio de actitud pudo estar influido por dos razones: 1 En primer lugar, el principio de acción estacionaria está ligado a la existencia de una función potencial, cuya existencia no requiere en el principio de d'Alembert. En segundo lugar, el principio de acción se presta a interpretaciones interpretaciones filosóficas y teleológicas que no le gustaban a Lagrange. Finalmente debe señalarse que el principio de d'Alembert es peculiarmente peculiarmente útil en la mecánica de sólidos donde puede usarse para plantear las ecuaciones de movimiento y cálculo de reacciones usando un campo de desplazamientos desplazamientos virtuales que sea diferenciable. En ese caso el cálculo mediante el principio de D'Alembert, que también se llama en ese contexto principio de los trabajos virtuales es ventajoso sobre el enfoque más simple de la mecánica newtoniana.