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Chapter 11: Problem 34

A barber鈥檚 chair with a person in it weighs 2100 N. The output plunger of a hydraulic system begins to lift the chair when the barber鈥檚 foot applies a force of 55 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 6.18.

Step by step solution

01

Understanding the Hydraulic System

In a hydraulic system, force is applied to a small piston and a larger force is exerted by a larger plunger due to Pascal's Principle. The ratio of forces is equal to the ratio of the areas of the pistons.
02

Setting Up the Force Equation

Given the force applied to the input piston is 55 N and the force exerted by the plunger is 2100 N. According to Pascal's Principle, \[ \frac{F_1}{F_2} = \frac{A_1}{A_2} \] 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.
03

Determining the Area Relationship

Express the area of a circle in terms of its radius: \[ A = \pi r^2 \] Using the area formula, the equation becomes \[ \frac{55}{2100} = \frac{\pi r_1^2}{\pi r_2^2} \] The \(\pi\) cancels out, simplifying to \[ \frac{55}{2100} = \frac{r_1^2}{r_2^2} \]
04

Solving for the Radius Ratio

Cross-multiply to solve the radius squared ratio: \[ 55 \times r_2^2 = 2100 \times r_1^2 \]Now, divide both sides by 55,\[ r_2^2 = \frac{2100}{55} \times r_1^2 \]Simplifying further gives: \[ r_2^2 = 38.18 \times r_1^2 \]Take the square root of both sides to find the ratio of the radii: \[ \frac{r_2}{r_1} = \sqrt{38.18} \]
05

Final Calculation

Calculate the square root to find the actual ratio: \[ \sqrt{38.18} \approx 6.18 \] Therefore, the ratio of the radius of the plunger to the radius of the piston is approximately 6.18.

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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 simple yet powerful mechanism that utilizes liquid pressure to multiply force. It operates on Pascal's Principle, which states that pressure exerted anywhere in a confined fluid is transmitted equally in all directions. This means that the pressure applied to one part of the fluid transfers to all other parts without diminishing.

In practical terms, a hydraulic system consists of two main components: the input piston and the output plunger. When you apply a force on the input piston, it creates pressure in the fluid, which then pushes on the larger output plunger. This arrangement allows for exerting a larger force than what was applied originally, which is why hydraulic systems are instrumental in various applications like car brakes and barber chairs.

It's important to remember that the ratio of the forces on the pistons/plunger is the same as the ratio of their areas. This unique property enables small forces to be converted into much larger ones without significant energy loss.
Force Ratio
Understanding the concept of force ratio is crucial in analyzing how a hydraulic system amplifies force. In our exercise, the force applied to the input piston (55 N) and the force exerted by the output plunger (2100 N) showcase the force ratio at work.

This ratio is described mathematically by Pascal's Principle, which can be expressed as:
  • \( \frac{F_1}{F_2} = \frac{A_1}{A_2} \)
Here, \( F_1 \) is the force applied to the input piston, and \( F_2 \) is the force exerted by the plunger. Meanwhile, \( A_1 \) and \( A_2 \) represent the areas of the input piston and the output plunger, respectively. The force ratio in any hydraulic system reflects the difference in surface areas of the pistons involved.

This is why the system can generate an output force considerably greater than the input force. By adjusting the size of the pistons and using a suitable fluid, engineers can design systems to amplify force in a controlled and efficient manner.
Area of a Circle
Understanding the area of a circle is indispensable when dealing with hydraulic systems, as the performance of these systems heavily relies on the piston's and plunger's area. The area calculation is straightforward: if you know the radius of a circle, you can find its area using the formula:
  • \( A = \pi r^2 \)
Here, \( A \) represents the area, \( \pi \) is a constant approximately equal to 3.14159, and \( r \) is the radius of the circle.

When you apply this concept to hydraulic systems, you see its significance. Since the force exerted is directly proportional to the area, larger pistons or plungers exert greater force. This happens because a larger surface area experiences more pressure, resulting in a higher output force.

For example, in our exercise, we determined the radius ratio by setting up the equation:
  • \( \frac{r_1^2}{r_2^2} = \frac{55}{2100} \)
This equation allowed us to solve for the relationship between the radius of the piston and the plunger, thus demonstrating how the area plays a pivotal role in the system's functionality. Understanding how to calculate and manipulate the area is essential for optimizing hydraulic systems for various applications.

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Most popular questions from this chapter

An underground pump initially forces water through a horizontal pipe at a flow rate of 740 gallons per minute. After several years of operation, corrosion and mineral deposits have reduced the inner radius of the pipe to 0.19 m from 0.24 m, but the pressure difference between the ends of the pipe is the same as it was initially. Find the final flow rate in the pipe in gallons per minute. Treat water as a viscous fluid.

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 incom- pressible fluid, find the length of a side of the square if the air speed within the ducts is \((\text { a } 3.0 \mathrm{m} / \mathrm{s} \text { and }(\mathrm{b}) 5.0 \mathrm{m} / \mathrm{s} \text { . }\)

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 m above that of the input piston?

An antifreeze solution is made by mixing ethylene glycol \(\left(\rho=1116 \mathrm{kg} / \mathrm{m}^{3}\right)\) with water. Suppose that the specific gravity of such a solution is \(1.0730 .\) Assuming that the total volume of the solution is the sum of its parts, determine the volume percentage of ethylene glycol in the solution.

Two hoses are connected to the same outlet using a Y-connector, as the drawing shows. The hoses A and B have the same length, but hose B has the larger radius. Each is open to the atmosphere at the end where the water exits. Water flows through both hoses as a viscous fluid, and Poiseuille鈥檚 law \(\left[Q=\pi R^{4}\left(P_{2}-P_{1}\right) /(8 \eta L)\right]\) applies to each. In this law, \(P_{2}\) is the pressure upstream, \(P_{1}\) is the pressure down- stream, and \(Q\) is the volume flow rate. The ratio of the radius of hose \(B\) to the radius of hose A is \(R_{B} / R_{A}=1.50 .\) Find the ratio of the speed of the water in hose \(\mathrm{B}\) to the speed in hose A.
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