The hydraulic system of a backhoe is used to lift a load as...

The hydraulic system of a backhoe is used to lift a load as shown in Figure $11.44 .$ (a) Calculate the force $F$ the slave cylinder must exert to support the $400-\mathrm{kg}$ load and the $150-\mathrm{kg}$ brace and shovel. (b) What is the pressure in the hydraulic fluid if the slave cylinder is $2.50 \mathrm{cm}$ in diameter? (c) What force would you have to exert on a lever with a mechanical advantage of 5.00 acting on a master cylinder $0.800 \mathrm{cm}$ in diameter to create this pressure?

Key Concepts

Pascal's Law

In hydraulics, Pascal's Law states that pressure applied at any point in a confined fluid is transmitted equally in all directions throughout the fluid. This principle is fundamental in hydraulic systems, allowing a small force applied on a small-area master cylinder to produce a larger force on a larger-area slave cylinder.

Force-Area Relationship

The relationship between force, pressure, and area (F = P × A) is a key concept in hydraulic systems. By knowing the area of a cylinder (which is calculated using its diameter) and the force required to support a load, one can determine the pressure within the hydraulic fluid and vice versa.

Hydraulic Force Multiplication

Hydraulic systems leverage the differences in cross-sectional areas of the master and slave cylinders to multiply force. A small input force on a small-area master cylinder can produce a much larger force on a larger-area slave cylinder, which is essential for tasks such as lifting heavy loads in machinery like backhoes.

Mechanical Advantage

Mechanical advantage is the factor by which a machine multiplies an input force. In the context of the hydraulic system and accompanying lever mechanism, the mechanical advantage of a lever reduces the required input force to achieve the desired output, effectively harnessing the force multiplication provided by both the lever and hydraulic system.

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