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Chapter 11: Problem 34

A barber's chair with a person in it weighs \(2100\) \(\mathrm{N}\). The output plunger of a hydraulic system begins to lift the chair when the barber's foot applies a force of \(55\) \(\mathrm{N}\) to the input piston. Neglect any height difference between the plunger and the piston. What is the ratio of the radius of the plunger to the radius of the piston?

Short Answer

Expert verified
The radius ratio of the plunger to the piston is approximately 6.157.

Step by step solution

01

Understanding Pascal's Principle

According to Pascal's principle, in a hydraulic system, the pressure applied at one point is transmitted undiminished throughout the fluid. This means the pressure (force divided by area) exerted on the plunger is the same as the pressure applied on the piston. Mathematically, it is expressed as \( \frac{F_1}{A_1} = \frac{F_2}{A_2} \), where \( F_1 \) and \( F_2 \) are the forces and \( A_1 \) and \( A_2 \) are the areas of the input piston and output plunger, respectively.
02

Setting Up the Equation

Given: - Force applied by the barber's foot (\( F_1 \)) = 55 N - Weight of the chair (\( F_2 \)) = 2100 N. Substitute these values into the equation: \[ \frac{55}{A_1} = \frac{2100}{A_2} \] Thus, \( A_2 = \frac{2100}{55} A_1 \).
03

Relating Area to Radius

The area \( A \) for a circular piston is given by the formula \( A = \pi r^2 \), where \( r \) is the radius. Thus, \( A_1 = \pi r_1^2 \) and \( A_2 = \pi r_2^2 \). Therefore, from the previous step: \[ \pi r_2^2 = \frac{2100}{55} \pi r_1^2 \] By cancelling out \( \pi \) from both sides, we obtain: \[ r_2^2 = \frac{2100}{55} r_1^2 \].
04

Finding the Radius Ratio

To find the ratio of the radii, take the square root of both sides: \[ \sqrt{r_2^2} = \sqrt{\frac{2100}{55} r_1^2} \] \[ r_2 = \sqrt{\frac{2100}{55}} r_1 \] Thus, the ratio \( \frac{r_2}{r_1} \) is \( \sqrt{\frac{2100}{55}} \) which simplifies to approximately 6.157.

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Key Concepts

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

Hydraulic System
A hydraulic system is a fascinating engineering marvel that uses fluid to transmit energy to perform work. At its core, it consists of two main components: the input piston and the output plunger. When a force is applied to the input piston, the hydraulic fluid transfers this force to the output plunger. This transfer is due to the incompressible nature of the fluid, ensuring that force is effectively moved from one place to another. This mechanism allows small applied forces to generate larger forces, making tasks like lifting heavy loads easier. Within a hydraulic system:
  • The hydraulic fluid plays a critical role in transferring forces.
  • The piston and plunger sizes affect how force is amplified.
  • The system follows Pascal's Principle for pressure transmission.
Understanding how these components work together helps us see why hydraulics are so integral in tools like lifts and heavy machinery.
Pressure Transmission
Pressure transmission in a hydraulic system is guided by Pascal's Principle. This principle states that when pressure is applied to a confined fluid, it is transmitted equally in all directions. This means that the pressure exerted on an input piston results in the same pressure being exerted on the output plunger.The key to this concept is understanding:
  • Pressure is calculated as the force applied per unit area \((P = \frac{F}{A})\).
  • In a hydraulic system, pressure remains constant throughout the fluid.
  • Thus, the force applied to a small area can result in a larger force at a larger area.
This principle allows hydraulic systems to lift heavy objects with minimal input force, leveraging the multiplication of force over larger areas.
Force and Area Relationship
The relationship between force and area is an important concept in understanding how hydraulic systems work. In these systems, it's crucial that the pressure applied (force per area) remains constant as it travels through the fluid. Therefore, the force exerted on the input piston and the corresponding force on the output plunger must direct to their respective areas.The area of the piston and plunger is related to their radii through the formula for the area of a circle, \(A = \pi r^2\). This helps define how:
  • The area changes with different radius sizes, influencing the force exerted.
  • When setup correctly, small input forces can generate substantial output forces.
  • Understanding the ratio of radii can determine the multiplication factor of forces in the system.
By manipulating the size and area of the piston's surfaces, hydraulic systems capitalize on these relationships to amplify input forces effectively.

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Most popular questions from this chapter

The atmospheric pressure above a swimming pool changes from 755 to \(765\) \(\mathrm{mm}\) of mercury. The bottom of the pool is a rectangle \((12 \mathrm{m} \times 24 \mathrm{m}) .\) By how much does the force on the bottom of the pool increase?

The density of ice is \(917 \mathrm{kg} / \mathrm{m}^{3}\), and the density of seawater is \(1025 \mathrm{kg} / \mathrm{m}^{3} .\) A swimming polar bear climbs onto a piece of floating ice that has a volume of \(5.2 \mathrm{m}^{3} .\) What is the weight of the heaviest bear that the ice can support without sinking completely beneath the water?

A solid cylinder (radius \(=0.150\) \(\mathrm{m}\), height \(=\) \(0.120\) \(\mathrm{m})\) has a mass of \(7.00\) \(\mathrm{kg} .\) This cylinder is floating in water. Then oil \(\left(\rho=725 \mathrm{kg} /\mathrm{m}^{3}\right)\) is poured on top of the water until the situation shown in the drawing results. How much of the height of the cylinder is in the oil?

A suitcase (mass \(m=16\) \(\mathrm{kg}\) ) is resting on the floor of an elevator. The part of the suitcase in contact with the floor measures \(0.50\) \(\mathrm{m} \times 0.15\) \(\mathrm{m}\). The elevator is moving upward with an acceleration of magnitude \(1.5 \mathrm{m} / \mathrm{s}^{2}\). What pressure (in excess of atmospheric pressure) is applied to the floor beneath the suitcase?

One of the concrete pillars that support a house is \(2.2 \mathrm{m}\) tall and has a radius of \(0.50 \mathrm{m}\). The density of concrete is about \(2.2 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}\). Find the weight of this pillar in pounds \((1 \mathrm{N}=0.2248 \mathrm{lb})\).
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