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Chapter 11: Problem 35

Multiple-Concept Example 8 presents an approach to problems of this kind. The hydraulic oil in a car lift has a density of \(8.30 \times 10^{2} \mathrm{kg} / \mathrm{m}^{3}\). The weight of the input piston is negligible. The radii of the input piston and output plunger are \(7.70 \times 10^{-3} \mathrm{m}\) and \(0.125 \mathrm{m},\) respectively. What input force \(F\) is needed to support the \(24500-\mathrm{N}\) combined weight of a car and the output plunger, when (a) the bottom surfaces of the piston and plunger are at the same level, and (b) the bottom surface of the output plunger is \(1.30 \mathrm{m}\) above that of the input piston?

Short Answer

Expert verified
(a) The input force needed is approximately 92.5 N; (b) with the height difference, it is approximately 72.9 N.

Step by step solution

01

Understand Pascal's Principle

Pascal's principle states that pressure applied to an enclosed fluid is transmitted undiminished to every part of the fluid and to the walls of its container. For this problem, it implies that the pressure exerted by the input piston is the same as the pressure at the output plunger.
02

Calculate the Pressure at the Input Piston

The input force on the piston, denoted as \(F_{in}\), applies pressure \(P = \frac{F_{in}}{A_{in}}\). The area of the input piston \(A_{in}\) is given by \(\pi r_{in}^2\), where \(r_{in} = 7.70 \times 10^{-3} \, \mathrm{m}\).
03

Calculate the Required Pressure at the Output Plunger

The output pressure must support the weight of the car and the plunger. Therefore, \(P_{out} = \frac{F_{out}}{A_{out}}\), where \(F_{out} = 24500 \, \mathrm{N}\) and \(A_{out} = \pi r_{out}^2\) with \(r_{out} = 0.125 \, \mathrm{m}\).
04

Equate the Pressures for Level Surfaces (Part a)

Set the pressure at the input equal to the pressure at the output: \(\frac{F_{in}}{A_{in}} = \frac{F_{out}}{A_{out}}\). Solve for \(F_{in}\): \[ F_{in} = F_{out} \times \frac{A_{in}}{A_{out}} = 24500 \, \mathrm{N} \times \frac{\pi (7.70 \times 10^{-3})^2}{\pi (0.125)^2} \].
05

Calculate the Input Force (Part a)

Compute \(F_{in}\) using the area formula: \(A_{in} = \pi (7.70 \times 10^{-3})^2 = 1.86 \times 10^{-4} \, \mathrm{m}^2\)\(A_{out} = \pi (0.125)^2 = 4.91 \times 10^{-2} \, \mathrm{m}^2\)Therefore, \[ F_{in} = 24500 \, \mathrm{N} \times \frac{1.86 \times 10^{-4}}{4.91 \times 10^{-2}} \approx 92.5 \, \mathrm{N} \].
06

Consider Height Difference (Part b)

When the plunger is 1.30 m above the piston, the pressure difference due to the height must be considered. The additional pressure exerted by the fluid column is \( \rho gh\), where \(\rho = 830 \, \mathrm{kg/m^3}\), \(g = 9.81 \, \mathrm{m/s^2}\), and \(h = 1.30 \, \mathrm{m}\).
07

Calculate Added Pressure and Adjusted Force (Part b)

Calculate the added pressure: \( \Delta P = 830 \, \mathrm{kg/m^3} \times 9.81 \, \mathrm{m/s^2} \times 1.30 \, \mathrm{m} = 10583.19 \, \mathrm{Pa}\).Thus, the input pressure must now equal \(P_{out} - \Delta P\). Solve \(\frac{F_{in,b}}{A_{in}} = \frac{F_{out}}{A_{out}} - \Delta P\). Adjust \(F_{in,b}\): \[ F_{in,b} = (P_{out} - \Delta P) \times A_{in} = \left(\frac{24500}{4.91 \times 10^{-2}} - 10583.19\right) \times 1.86 \times 10^{-4} \approx 72.9 \, \mathrm{N} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hydraulic Systems
Hydraulic systems are fascinating because they use fluid power for various applications, from car lifts to heavy machinery. A basic hydraulic system consists of two pistons--an input and an output--connected by a sealed system filled with hydraulic fluid.

The input piston is where force is initially applied, which in a car lift scenario, might be from a pump operating manually or by a motor. The magic of hydraulics lies in how small input forces can generate large output forces to lift heavy objects.
  • Force applied on the input piston is transferred through the fluid to the output piston.
  • The fluid in the system is almost incompressible, maintaining pressure across the system.
Pascal's Principle is at play here, ensuring that any change in pressure applied to the enclosed fluid is evenly distributed throughout.

This means the pressure is constant from one part of the system to another, allowing a small force exerted on a small area to induce a larger force over a larger area. It's the principle behind the power of hydraulic lifts: using the pressure consistency in the fluid to multiply the force.
Pressure Calculations
Pressure in hydraulic systems is defined as the force exerted per unit area on the surface of the system. In these systems, the understanding of pressure calculations is crucial to determine the amount of force needed to lift or hold an object.

For hydraulic lifts, we derive pressure calculations from the basic formula:\[P = \frac{F}{A}\]
Here, \(P\) is the pressure, \(F\) is the force applied, and \(A\) is the area over which the force is distributed.
  • The input force results in a pressure applied to the hydraulic fluid.
  • This pressure is transmitted through the fluid to the output piston, which does the lifting work.
Calculating the required force involves assessing both the areas of the pistons and the weight of the load.
The pressure at the output is required to balance the gravitational force acting on the load. When a height difference is present in the system, an additional calculation factor is added--the gravitational pull on the fluid column, which adjusts the required input force accordingly.
Fluid Mechanics
Fluid mechanics is the study of fluids (liquids and gases) in motion or at rest. It is a key component in understanding how hydraulic systems operate. Fluid mechanics principles explain how pressure is maintained in a fluid and how it transmits force.

Key concepts include:
  • Density, which plays a significant role in calculating pressure changes due to height differences in a hydraulic system, as seen in the adjustment calculations when the output plunger is raised.
  • Incompressibility, which assumes that the fluid does not change in volume under pressure, central to efficient hydraulic systems.
  • Continuity, which implies that mass is conserved as fluid moves through the system, ensuring consistent pressure.
Hydraulic fluids, which are primarily oil-based in car lifts, help maintain low friction while transferring energy effectively across the system, allowing the hydraulic lift to operate smoothly and efficiently.

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Most popular questions from this chapter

The Mariana trench is located in the floor of the Pacific Ocean at a depth of about \(11000 \mathrm{m}\) below the surface of the water. The density of seawater is \(1025 \mathrm{kg} / \mathrm{m}^{3} .\) (a) If an underwater vehicle were to explore such a depth, what force would the water exert on the vehicle's observation window (radius \(=0.10 \mathrm{m}) ?\) (b) For comparison, determine the weight of a jetliner whose mass is \(1.2 \times 10^{5} \mathrm{kg}.\)

The vertical surface of a reservoir dam that is in contact with the water is \(120 \mathrm{m}\) wide and \(12 \mathrm{m}\) high. The air pressure is one atmosphere. Find the magnitude of the total force acting on this surface in a completely filled reservoir. (Hint: The pressure varies linearly with depth, so you must use an average pressure. )

A water line with an internal radius of \(6.5 \times 10^{-3} \mathrm{m}\) is connected to a shower head that has 12 holes. The speed of the water in the line is \(1.2 \mathrm{m} / \mathrm{s}\). (a) What is the volume flow rate in the line? (b) At what speed does the water leave one of the holes (effective hole radius \(=4.6 \times 10^{-4} \mathrm{m}\) ) in the head?

A blood transfusion is being set up in an emergency room for an accident victim. Blood has a density of \(1060 \mathrm{kg} / \mathrm{m}^{3}\) and a viscosity of \(4.0 \times 10^{-3} \mathrm{Pa} \cdot \mathrm{s} .\) The needle being used has a length of \(3.0 \mathrm{cm}\) and an inner radius of \(0.25 \mathrm{mm} .\) The doctor wishes to use a volume flow rate through the needle of \(4.5 \times 10^{-8} \mathrm{m}^{3} / \mathrm{s} .\) What is the distance \(h\) above the victim's arm where the level of the blood in the transfusion bottle should be located? As an approximation, assume that the level of the blood in the transfusion bottle and the point where the needle enters the vein in the arm have the same pressure of one atmosphere. (In reality, the pressure in the vein is slightly above atmospheric pressure.)

A room has a volume of \(120 \mathrm{m}^{3} .\) An air-conditioning system is to replace the air in this room every twenty minutes, using ducts that have a square cross section. Assuming that air can be treated as an incompressible fluid, find the length of a side of the square if the air speed within the ducts is (a) \(3.0 \mathrm{m} / \mathrm{s}\) and (b) \(5.0 \mathrm{m} / \mathrm{s}\)
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