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PROBLEMA Suponga que el agua sale de un depósito por un orificio circular de área Ak en su fondo. Cuando el agua sale por el orif orific icio io,, la fric fricci ción ón y la con contrac tracci ción ón de la corri orrien ente te cer cerca del orificio reducen el volumen de agua que sale del depósito.. Determine la ecuación diferencial para la altura h del agua en el instante t para el depósito que se muestra a continuación. El radio del orificio es de 2 pulg
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Solución El volumen del agua en el tanque en el instante instante t es V w = Ah Con esa ecuación podemos plantear una diferencial entre la altura y el tiempo en el que disminuye el volumen de agua en el recipiente:
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•
Usando
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sustituyendo estos valores para las condiciones establecidas:
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Hallar el tiempo en vaciar el cono de Radio “R” “ R” en la base con 3 agujeros de area “a” como se muestra en la figura
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Problema: Se tiene un recipiente recipiente recto recto cuya cuya sección transver transversal sal es como como semielse de
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Es decir estará a la mitad, alrededor de las 12h + 20min + 55seg
≈ 12h 21min.
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PROBLEMA 12) Un tanque rectangular de base
y altura H tiene dos agujeros de
área A, uno en el fondo y otro en una cara lateral lateral a la mitad de la altura altura H. Si el tanque está inicialmente lleno de agua y con un coeficiente de gasto c. Durante cuanto tiempo saldrá agua por el agujero lateral.
SOLUCIÓN: Por Torricelli, como hay dos agujeros:
2ℎ + cA 2 ℎ Por otro lado: dV = -A(h)dh Pero A(h) = . En (2): dV = - dh
dV = cA
Ahora (3) en (1):
. . .(1) . . .(2) . . .(3)
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17)Dos depósitos pequeños, con orificios e idénticos en el fondo se vacían al mismo tiempo. Uno de ellos es un cilindro con el eje vertical y el otro, un cono con el vértice hacia abajo. Si tienen bases iguales y la altura del cilindro es H. ¿ Cuál es la altura del cono ? Considerar el mismo coeficiente de descarga para ambos.
•
SOLUCION : Para el cilindro: Sean V(t) y h(t) el volumen de agua en el
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