/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 34 A barber's chair with a person i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 34

A barber's chair with a person in it weighs 2100 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 is approximately 6.18.

Step by step solution

01

Understand the Problem

We are given that the combined weight of a barber's chair and the person sitting in it is 2100 N. This weight is lifted by a hydraulic system when a force of 55 N is applied to the input piston by the barber's foot. We need to find the ratio of the radius of the plunger to the radius of the piston.
02

Apply Pascal's Principle

According to Pascal's principle for hydraulic systems, the pressure applied to the input piston is transmitted equally throughout the fluid. This results in an output force on the plunger. Thus, the input and output forces are related by the pressures: \[ \text{Input Pressure} = \text{Output Pressure} \].This can be expressed as: \[ \frac{F_1}{A_1} = \frac{F_2}{A_2} \]where \(F_1 = 55\, \text{N}\), \(A_1\) is the area of the input piston, \(F_2 = 2100\, \text{N}\), and \(A_2\) is the area of the plunger.
03

Express Areas in terms of Radii

The area \(A\) of a circle is given by \(\pi r^2\), where \(r\) is the radius. Thus, we can write:\[ A_1 = \pi r_1^2 \quad \text{and} \quad A_2 = \pi r_2^2 \]where \(r_1\) and \(r_2\) are the radii of the piston and plunger, respectively.
04

Solve for the Radius Ratio

Substitute the expressions for \(A_1\) and \(A_2\) into the pressure equation:\[ \frac{F_1}{\pi r_1^2} = \frac{F_2}{\pi r_2^2} \]Cancel \(\pi\) from both sides to get:\[ \frac{F_1}{r_1^2} = \frac{F_2}{r_2^2} \]Reorganizing gives:\[ \left( \frac{r_2}{r_1} \right)^2 = \frac{F_2}{F_1} \]
05

Calculate the Ratio

Substitute the forces into the equation:\[ \left( \frac{r_2}{r_1} \right)^2 = \frac{2100}{55} \]Calculate the right-hand side:\[ \left( \frac{r_2}{r_1} \right)^2 = 38.1818 \]Taking the square root of both sides gives:\[ \frac{r_2}{r_1} = \sqrt{38.1818} \approx 6.18 \]
06

Conclude the Solution

The ratio of the radius of the plunger to the radius of the piston is approximately 6.18.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Pascal's Principle
Pascal's Principle is a fundamental concept in fluid mechanics. It states that when pressure is applied to a confined fluid, it is transmitted uniformly throughout the fluid in all directions. This is crucial in hydraulic systems, such as the one in a barber's chair lifting mechanism. When force is applied to the input piston, the pressure generated is equal across the entire fluid medium. This pressure then acts on the plunger, allowing it to lift an object, like a chair.
With Pascal's Principle, hydraulic systems can effectively amplify force. This is why a small force applied at one point results in a larger force at another point. This principle is used in many applications like car brakes and machinery, demonstrating its extensive utility.
Pressure
Pressure is defined as the force applied per unit area. In a hydraulic system, the pressure must be the same at the input and output due to Pascal's Principle. This is expressed mathematically as:
  • \( \text{Pressure} = \frac{\text{Force}}{\text{Area}} \)
Using this relationship, equal pressure across the system can be described by: \( \frac{F_1}{A_1} = \frac{F_2}{A_2} \), where subscript 1 refers to the input piston and subscript 2 to the output plunger.
Forces applied in hydraulic systems often cause significant pressure changes that directly influence the movement and force exerted by the system components. Therefore, understanding pressure gives insight into how equipment, like a barbershop chair, operates efficiently with minimal force input.
Force
The concept of force is integral to understanding hydraulic systems. Force in such systems is often used to initiate movement or lift heavy objects. In our given problem, an initial force of 55 N applied at the piston results in a much larger force of 2100 N being exerted by the plunger.
The relationship between the forces applied at different points in a hydraulic system can be described by:
  • \( \frac{F_1}{\pi r_1^2} = \frac{F_2}{\pi r_2^2} \)
This equation shows that the input and output forces depend on the radii of the respective pistons and plungers. Hydraulic systems leverage this force difference to produce large output forces from small input forces. This capability is what makes hydraulic machines indispensable in lifting and other applications requiring substantial force.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Prairie dogs are burrowing rodents. They do not suffocate in their burrows, because the effect of air speed on pressure creates sufficient air circulation. The animals maintain a difference in the shapes of two entrances to the burrow, and because of this difference, the air \(\left(\rho=1.29 \mathrm{kg} / \mathrm{m}^{3}\right)\) blows past the openings at different speeds, as the drawing indicates. Assuming that the openings are at the same vertical level, find the difference in air pressure between the openings and indicate which way the air circulates.

An airplane wing is designed so that the speed of the air across the top of the wing is \(251 \mathrm{m} / \mathrm{s}\) when the speed of the air below the wing is \(225 \mathrm{m} / \mathrm{s} .\) The density of the air is \(1.29 \mathrm{kg} / \mathrm{m}^{3} .\) What is the lifting force on a wing of area \(24.0 \mathrm{m}^{2} ?\)

(a) The mass and the radius of the sun are, respectively, \(1.99 \times\) \(10^{30} \mathrm{kg}\) and \(6.96 \times 10^{8} \mathrm{m} .\) What is its density? (b) If a solid object is made from a material that has the same density as the sun, would it sink or float in water? Why? (c) Would a solid object sink or float in water if it were made from a material whose density was the same as that of the planet Saturn (mass \(=5.7 \times 10^{26} \mathrm{kg},\) radius \(\left.=6.0 \times 10^{7} \mathrm{m}\right) ?\) Provide a reason for your answer.

A ship is floating on a lake. Its hold is the interior space beneath its deck; the hold is empty and is open to the atmosphere. The hull has a hole in it, which is below the water line, so water leaks into the hold. The effective area of the hole is \(8.0 \times 10^{-3} \mathrm{m}^{2}\) and is located \(2.0 \mathrm{m}\) beneath the surface of the lake. What volume of water per second leaks into the ship?

The blood speed in a normal segment of a horizontal artery is \(0.11 \mathrm{m} / \mathrm{s} .\) An abnormal segment of the artery is narrowed down by an arteriosclerotic plaque to one-fourth the normal cross-sectional area. What is the difference in blood pressures between the normal and constricted segments of the artery?
See all solutions

Recommended explanations on Physics Textbooks

Oscillations

Read Explanation

Thermodynamics

Read Explanation

Atoms and Radioactivity

Read Explanation

Mechanics and Materials

Read Explanation

Physical Quantities And Units

Read Explanation

Fundamentals of Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.