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1. Hydraulic jacks are one of the many tools humans use to develop a so-called "mechanical advantage". The basic idea is that a relatively small force applied to the smaller-area piston can counteract a relatively large force applied to the larger-area piston. F₂ a) Ignoring the weight of the fluid (i.e., ignoring changes in hydrostatic pressure due to the density of the fluid) and any friction between the piston and the sidewalls, show that F₂ A₁ F₁ in static equilibrium. b) The mechanical advantage of the hydraulic jack can be defined as the ratio of the force applied to piston 2 to the force applied to piston 1 in order to keep the system in static equilibrium. For circular pistons, use the relationship in part a) to plot the mechanical advantage as a function of the diameter ratio of the pistons. Explain the significance of this mechanical advantage. c) Consider a hydraulic jack in static equilibrium with both pistons at the same elevation. If you do not consider the weight of the fluid in your analysis (i.e., if you do not consider hydrostatic pressure changes due to elevation differences in the fluid), what would happen if the force on piston 2 were increased slightly? d) Your answer to part c) should indicate that the weight of the fluid must be accounted for. If the two pistons are initially in equilibrium at the same elevation and the force on piston 2 is increased by AF2, derive an expression for the new elevation difference between the pistons, Ah = f(AF2, A2, 7), where y is the specific weight of the hydraulic fluid.

1. Hydraulic jacks are one of the many tools humans use to develop a so-called "mechanical advantage". The basic idea is that a relatively small force applied to the smaller-area piston can counteract a relatively large force applied to the larger-area piston. F₂ a) Ignoring the weight of the fluid (i.e., ignoring changes in hydrostatic pressure due to the density of the fluid) and any friction between the piston and the sidewalls, show that F₂ A₁ F₁ in static equilibrium. b) The mechanical advantage of the hydraulic jack can be defined as the ratio of the force applied to piston 2 to the force applied to piston 1 in order to keep the system in static equilibrium. For circular pistons, use the relationship in part a) to plot the mechanical advantage as a function of the diameter ratio of the pistons. Explain the significance of this mechanical advantage. c) Consider a hydraulic jack in static equilibrium with both pistons at the same elevation. If you do not consider the weight of the fluid in your analysis (i.e., if you do not consider hydrostatic pressure changes due to elevation differences in the fluid), what would happen if the force on piston 2 were increased slightly? d) Your answer to part c) should indicate that the weight of the fluid must be accounted for. If the two pistons are initially in equilibrium at the same elevation and the force on piston 2 is increased by AF2, derive an expression for the new elevation difference between the pistons, Ah = f(AF2, A2, 7), where y is the specific weight of the hydraulic fluid.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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1. Hydraulic jacks are one of the many tools humans use to develop a so-called "mechanical advantage". The basic idea is that a relatively small force applied to the smaller-area piston can counteract a relatively large force applied to the larger-area piston. F₂ a) Ignoring the weight of the fluid (i.e., ignoring changes in hydrostatic pressure due to the density of the fluid) and any friction between the piston and the sidewalls, show that F2 = F₁ AR A₁ in static equilibrium. b) The mechanical advantage of the hydraulic jack can be defined as the ratio of the force applied to piston 2 to the force applied to piston 1 in order to keep the system in static equilibrium. For circular pistons, use the relationship in part a) to plot the mechanical advantage as a function of the diameter ratio of the pistons. Explain the significance of this mechanical advantage. c) Consider a hydraulic jack in static equilibrium with both pistons at the same elevation. If you do not consider the weight of the fluid in your analysis (i.e., if you do not consider hydrostatic pressure changes due to elevation differences in the fluid), what would happen if the force on piston 2 were increased slightly? d) Your answer to part c) should indicate that the weight of the fluid must be accounted for. If the two pistons are initially in equilibrium at the same elevation and the force on piston 2 is increased by AF2, derive an expression for the new elevation difference between the pistons, Ah = f(AF2, A2, 7), where y is the specific weight of the hydraulic fluid.
Transcribed Image Text:1. Hydraulic jacks are one of the many tools humans use to develop a so-called "mechanical advantage". The basic idea is that a relatively small force applied to the smaller-area piston can counteract a relatively large force applied to the larger-area piston. F₂ a) Ignoring the weight of the fluid (i.e., ignoring changes in hydrostatic pressure due to the density of the fluid) and any friction between the piston and the sidewalls, show that F2 = F₁ AR A₁ in static equilibrium. b) The mechanical advantage of the hydraulic jack can be defined as the ratio of the force applied to piston 2 to the force applied to piston 1 in order to keep the system in static equilibrium. For circular pistons, use the relationship in part a) to plot the mechanical advantage as a function of the diameter ratio of the pistons. Explain the significance of this mechanical advantage. c) Consider a hydraulic jack in static equilibrium with both pistons at the same elevation. If you do not consider the weight of the fluid in your analysis (i.e., if you do not consider hydrostatic pressure changes due to elevation differences in the fluid), what would happen if the force on piston 2 were increased slightly? d) Your answer to part c) should indicate that the weight of the fluid must be accounted for. If the two pistons are initially in equilibrium at the same elevation and the force on piston 2 is increased by AF2, derive an expression for the new elevation difference between the pistons, Ah = f(AF2, A2, 7), where y is the specific weight of the hydraulic fluid.
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