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Chapter 11: Problem 39

The drawing shows a hydraulic system used with disc brakes. The force \(\overrightarrow{\mathbf{F}}\) is applied perpendicularly to the brake pedal. The pedal rotates about the axis shown in the drawing and causes a force to be applied perpendicularly to the input piston (radius \(=9.50 \times 10^{-3} \mathrm{m} )\) in the master cylinder. The resulting pressure is transmitted by the brake fluid to the output plungers (radii \(=1.90 \times 10^{-2} \mathrm{m}\) ), which are covered with the brake linings. The linings are pressed against both sides of a disc attached to the rotating wheel. Suppose that the magnitude of \(\overrightarrow{\mathbf{F}}\) is 9.00 \(\mathrm{N}\) . Assume that the input piston and the output plungers are at the same vertical level, and find the force applied to each side of the rotating disc.

Short Answer

Expert verified
The force applied to each side of the disc is approximately 90.0 N.

Step by step solution

01

Calculate the Area of the Input Piston

The area of the input piston is calculated using the formula for the area of a circle, \( A = \pi r^2 \). Here, the radius \( r \) of the input piston is given as \( 9.50 \times 10^{-3} \text{ m} \). Thus, the area is \( A = \pi (9.50 \times 10^{-3})^2 \).
02

Calculate the Pressure in the Hydraulic System

The pressure \( P \) exerted by the force \( F \) on the input piston can be calculated using the formula \( P = \frac{F}{A} \), where \( F = 9.00 \text{ N} \) is the force applied on the brake pedal. Substitute the value of the area from Step 1 to find the pressure.
03

Calculate the Area of the Output Plungers

The area of the output plungers is also circular and is found using the same formula, \( A = \pi r^2 \). Here, the radius given is \( 1.90 \times 10^{-2} \text{ m} \). Therefore, the area is \( A = \pi (1.90 \times 10^{-2})^2 \).
04

Calculate the Force on the Output Plungers

The force \( F_{output} \) exerted by the output plungers is found by multiplying the pressure from Step 2 by the area of the output plungers from Step 3, using the formula \( F = PA \). This will give the total force applied by each plunger.
05

Calculate the Force on Each Side of the Disc

Since there are two output plungers (one for each side of the disc), the force calculated in Step 4 is distributed between them equally. Thus, the force on each side of the disc is \( \frac{F_{output}}{2} \). Compute this to find the force applied to each side of the disc.

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Key Concepts

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

Pressure Calculation
In a hydraulic system, understanding pressure is key. Pressure is the force exerted per unit area on an object. Imagine you press your hand against a wall. The amount of force you apply is spread over the area of your hand. This principle is similar in hydraulic systems.
To calculate pressure in such a system, we use the formula: \[ P = \frac{F}{A} \] Where:
  • \( P \) is the pressure
  • \( F \) is the force applied
  • \( A \) is the area over which the force is distributed
In our example, the force (\( F \)) applied to the brake pedal is 9.00 N. The pressure created by this force is distributed evenly by the hydraulic fluid within the closed system. Remember, in hydraulics, pressure is generally the same throughout the fluid. Knowing this allows us to find the force exerted elsewhere in the system.
Force Distribution
Force distribution is crucial in a hydraulic system because it allows for efficient transfer of power. Once pressure is calculated, it is evenly distributed across the system through the hydraulic fluid. Think of it like spokes in a wheel, each part relies on another to maintain balance.
When we talk about force distribution in hydraulics specifically, the focus usually falls on how force applied to one piston affects the others. This balance is what makes hydraulics so powerful and efficient.
In the exercise context, the initial force applied to the brake pedal eventually influences the brake linings that apply stopping force on the wheel's disc. The role of pressure becomes evident here鈥攊t acts as the bridge which helps distribute the force from one smaller area (input piston) to a larger area (output plungers). The efficient spreading of force ensures each part of the system receives the necessary power level to function correctly.
Circular Area Calculation
Understanding how to calculate the area of a circle is fundamental in analyzing hydraulic systems, especially because we deal with circular pistons and plungers. The equation used to calculate the circular area is: \[ A = \pi r^2 \] Where:
  • \( A \) is the area
  • \( r \) is the radius of the circle
In this exercise, we calculate two different circular areas; one for the input piston and the other for the output plungers.
Using the radii provided (\(9.50 \times 10^{-3} \, \text{m} \) for the input piston and \(1.90 \times 10^{-2} \, \text{m} \) for the output plungers), we find these respective areas using the formula above. By knowing each area, we can effectively determine how the force exerted on each piston affects the rest of the system.
These calculations are foundational for understanding how variations in piston size contribute to the force applied on the brake system's disc, demonstrating the elegance and efficiency of hydraulic systems.

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Most popular questions from this chapter

A mercury barometer reads 747.0 mm on the roof of a building and 760.0 \(\mathrm{mm}\) on the ground. Assuming a constant value of 1.29 \(\mathrm{kg} / \mathrm{m}^{3}\) for the density of air, determine the height of the building.

A siphon tube is useful for removing liquid from a tank. The siphon tube is first filled with liquid, and then one end is inserted into the tank. Liquid then drains out the other end, as the drawing illustrates. (a) Using reasoning similar to that employed in obtaining Torricelli鈥檚 theorem (see Example 16), derive an expression for the speed \(v\) of the fluid emerging from the tube. This expression should give \(v\) in terms of the vertical height \(y\) and the acceleration due to gravity \(g\) . (Note that this speed does not depend on the depth \(d\) of the tube below the surface of the liquid.) (b) At what value of the vertical distance y will the siphon stop working? (c) Derive an expression for the absolute pressure at the highest point in the siphon (point \(A )\) in terms of the atmospheric pressure \(P_{0},\) the fluid density \(\rho, g,\) and the heights \(h\) and \(y\) (Note that the fluid speed at point \(A\) is the same as the speed of the fluid emerging from the tube, because the cross-sectional area of the tube is the same everywhere.)

A cylinder is fitted with a piston, beneath which is a spring, as in the drawing. The cylinder is open to the air at the top. Friction is absent. The spring constant of the spring is 3600 N/m. The piston has a negligible mass and a radius of 0.024 m. (a) When the air beneath the piston is completely pumped out, how much does the atmospheric pressure cause the spring to compress? (b) How much work does the atmospheric pressure do in compressing the spring?

A \(58-\mathrm{kg}\) skier is going down a slope oriented \(35^{\circ}\) above the horizontal. The area of each ski in contact with the snow is 0.13 \(\mathrm{m}^{2}\) Determine the pressure that each ski exerts on the snow.

A person who weighs 625 \(\mathrm{N}\) is riding a \(98-\mathrm{N}\) mountain bike. Suppose that the entire weight of the rider and bike is supported equally by the two tires. If the pressure in each tire is \(7.60 \times 10^{5} \mathrm{Pa},\) what is the area of contact between each tire and the ground?
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