ENUNCIADO EJEMPLO 13

    Disco homogéneo de masa M y radio R que rueda sin deslizar sobre el plano horizontal y gira alrededor del eje vertical a una distancia d. Sobre su borde se mueve una partícula de masa m.

   Paso 0. Reiniciación de las variables del sistema y llamada a los paquetes linalg, plots y plottools.

> restart;

> with(linalg):with(plots):with(plottools):

Warning, the protected names norm and trace have been redefined and unprotected

Warning, the name changecoords has been redefined

Warning, the name arrow has been redefined

> libname:="C:/",libname:

> with(mecapac3d):

   Paso 1. Definimos las coordenadas generalizadas del sistema en una lista que se denominará cg.

> cg:=[alpha,theta]:

   Paso 2. Definición mediante variables de los elementos que forman el sistema mecánico. Es decir, el aro, el disco y el muelle.

> rotdisc:=evalm(rota(d/R*alpha,1) &*rota(Pi/2,2)):

> d1:=[disco,[0,0,0],rotdisc,M,R]:

> m1:=[punto,0,R*cos(theta),R*sin(theta),m]:

   Paso 3. Definición de los elementos gráficos que definiran nuestro sistema de ejes.

> ejeX:=[vector,[0,0,0],[10,0,0],red]:

> ejeY:=[vector,[0,0,0],[0,10,0],green]:

> ejeZ:=[vector,[0,0,0],[0,0,10],blue]:

> TO := [texto,[0,0,-1],"O"]:

> TX := [texto,[10,0,1],"X"]:

> TY := [texto,[0,10,1],"Y"]:

> TZ := [texto,[0,0,11],"Z"]:

   Paso 4. Definición de la variable sistema que agrupa en una lista todos los elementos anteriores.

> cgsub:=[d*cos(alpha),d*sin(alpha),R]:

> rotsub:=rota(alpha,3):

> s1:=[subsistema2,cgsub,rotsub,[d1,m1]]:

> sistema:=[s1,ejeX,ejeY,ejeZ,TO,TX,TY,TZ]:

   Paso 5. Obtención de la energía cinética del sistema mediante fT asignándola a la variable T.

> T:=fT(sistema);

T := 135/2*sin(alpha)^2*alpha1^2+135/2*cos(alpha)^2*alpha1^2+11.71875000*cos(1.200000000*alpha)^2*alpha1^2+11.71875000*sin(1.200000000*alpha)^2*alpha1^2+5/2*(-3*sin(alpha)*alpha1-2.5*cos(alpha)*alpha1...T := 135/2*sin(alpha)^2*alpha1^2+135/2*cos(alpha)^2*alpha1^2+11.71875000*cos(1.200000000*alpha)^2*alpha1^2+11.71875000*sin(1.200000000*alpha)^2*alpha1^2+5/2*(-3*sin(alpha)*alpha1-2.5*cos(alpha)*alpha1...

   Paso 6. Obtención de la energía potencial del sistema mediante fV asignándola a la variable V.

> V:=fV(sistema);

V := 490.00+122.50*sin(theta)

   Paso 7. Obtención de la lagrangiana como diferencia de energías entre la energía cinética y la potencial.

> L:=T-V:

   Paso 8. Obtención de las ecuaciones de lagrange para las dos coordenadas generalizadas mediante el operador Ec_lag

> ecua:=ec_lag();

ecua := [135*sin(alpha(t))^2*diff(alpha(t), `$`(t, 2))+135*cos(alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*cos(1.200000000*alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*sin(1.200000000*alpha...ecua := [135*sin(alpha(t))^2*diff(alpha(t), `$`(t, 2))+135*cos(alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*cos(1.200000000*alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*sin(1.200000000*alpha...ecua := [135*sin(alpha(t))^2*diff(alpha(t), `$`(t, 2))+135*cos(alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*cos(1.200000000*alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*sin(1.200000000*alpha...ecua := [135*sin(alpha(t))^2*diff(alpha(t), `$`(t, 2))+135*cos(alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*cos(1.200000000*alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*sin(1.200000000*alpha...ecua := [135*sin(alpha(t))^2*diff(alpha(t), `$`(t, 2))+135*cos(alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*cos(1.200000000*alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*sin(1.200000000*alpha...ecua := [135*sin(alpha(t))^2*diff(alpha(t), `$`(t, 2))+135*cos(alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*cos(1.200000000*alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*sin(1.200000000*alpha...ecua := [135*sin(alpha(t))^2*diff(alpha(t), `$`(t, 2))+135*cos(alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*cos(1.200000000*alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*sin(1.200000000*alpha...ecua := [135*sin(alpha(t))^2*diff(alpha(t), `$`(t, 2))+135*cos(alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*cos(1.200000000*alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*sin(1.200000000*alpha...ecua := [135*sin(alpha(t))^2*diff(alpha(t), `$`(t, 2))+135*cos(alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*cos(1.200000000*alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*sin(1.200000000*alpha...ecua := [135*sin(alpha(t))^2*diff(alpha(t), `$`(t, 2))+135*cos(alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*cos(1.200000000*alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*sin(1.200000000*alpha...ecua := [135*sin(alpha(t))^2*diff(alpha(t), `$`(t, 2))+135*cos(alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*cos(1.200000000*alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*sin(1.200000000*alpha...ecua := [135*sin(alpha(t))^2*diff(alpha(t), `$`(t, 2))+135*cos(alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*cos(1.200000000*alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*sin(1.200000000*alpha...ecua := [135*sin(alpha(t))^2*diff(alpha(t), `$`(t, 2))+135*cos(alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*cos(1.200000000*alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*sin(1.200000000*alpha...ecua := [135*sin(alpha(t))^2*diff(alpha(t), `$`(t, 2))+135*cos(alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*cos(1.200000000*alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*sin(1.200000000*alpha...ecua := [135*sin(alpha(t))^2*diff(alpha(t), `$`(t, 2))+135*cos(alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*cos(1.200000000*alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*sin(1.200000000*alpha...ecua := [135*sin(alpha(t))^2*diff(alpha(t), `$`(t, 2))+135*cos(alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*cos(1.200000000*alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*sin(1.200000000*alpha...ecua := [135*sin(alpha(t))^2*diff(alpha(t), `$`(t, 2))+135*cos(alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*cos(1.200000000*alpha(t))^2*diff(alpha(t), `$`(t, 2))+23.43750000*sin(1.200000000*alpha...

   Paso 9. Asignación de valores numéricos a los parámetros que queden sun asignar para poder proceder a la integración numérica.

> g:=9.8:M:=15:m:=5:R:=2.5:d:=3:

   Paso 10. Integración numérica del problema mediante la función fint asignando el resultado a la variable res.

> res:=fint([0,1,0,0]):

   Paso 11. Procedemos a realizar una animación del movimiento del conjunto por medio de la función dibu3.

> dibu3(2,50);

[Plot]

>