En la tienda de mascotas "Animalo-T", se desea elevar un elefante de 2,900 kg utilizando una elevadora hidráulica de plato grande circular de 75 cm de radio y plato pequeño circular de 13 cm de radio. ¿Calcula cuánta fuerza hay que hacer en el émbolo pequeño para elevar al paquidermo?

Answer:

Se requiere una fuerza de 854.473 newtons sobre el émbolo pequeño.

Step-by-step explanation:

Por el Principio de Pascal se conoce que el esfuerzo experimentado por el elefante es igual a la presión ejercida por el plato pequeño. Es decir:

[tex]\frac{F_{1}}{A_{1}} = \frac{F_{2}}{A_{2}}[/tex] (1)

Donde:

[tex]F_{1}[/tex] - Fuerza experimentada por el elefante, medida en newtons.

[tex]F_{2}[/tex] - Fuerza aplicada sobre el plato pequeño, medida en newtons.

[tex]A_{1}[/tex] - Área del plato grande, medida en metros cuadrados.

[tex]A_{2}[/tex] - Área del plato pequeño, medida en metros cuadrados.

La fuerza aplicada sobre el plato pequeño es:

[tex]F_{2} = \left(\frac{A_{2}}{A_{1}} \right)\cdot F_{1}[/tex]

La fuerza experimentada por el elefante es su propio peso. Por otra parte, el área del plato es directamente proporcional al cuadrado de su diámetro. Es decir:

[tex]F_{2} = \left(\frac{D_{2}}{D_{1}} \right)^{2}\cdot m\cdot g[/tex] (2)

Donde:

[tex]D_{1}[/tex] - Diámetro del plato grande, medido en centímetros.

[tex]D_{2}[/tex] - Diámetro del plato pequeño, medido en centímetros.

[tex]m[/tex] - Masa del elefante, medida en kilogramos.

[tex]g[/tex] - Aceleración gravitacional, medida en metros por segundo cuadrado.

Si sabemos que [tex]D_{1} = 0.75\,m[/tex], [tex]D_{2} = 0.13\,m[/tex], [tex]m = 2900\,kg[/tex] y [tex]g = 9.807\,\frac{m}{s^{2}}[/tex], entonces la fuerza a aplicar al émbolo pequeño es:

[tex]F_{2} = \left(\frac{0.13\,m}{0.75\,m} \right)^{2}\cdot (2900\,kg)\cdot \left(9.807\,\frac{m}{s^{2}} \right)[/tex]

[tex]F_{2} = 854.473\,N[/tex]

Se requiere una fuerza de 854.473 newtons sobre el émbolo pequeño.

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