A garage lift has a 45 -cm-diameter piston supporting the...

A garage lift has a 45 -cm-diameter piston supporting the load. Compressed air with maximum pressure $500 \mathrm{kPa}$ is applied to a small piston at the other end of the hydraulic system. What's the maximum mass the lift can support?

Key Concepts

Circular Area Calculation

When dealing with pistons or any circular interface, the area is calculated using the formula A = ?r², where r is the radius of the circle. This calculation is critical in determining the force generated on a circular piston given a certain pressure, as it directly influences the amount of force that can be exerted for lifting or supporting loads.

Pascal's Principle

This principle states that any change in pressure applied to a confined fluid is transmitted undiminished throughout the fluid in all directions. It is the fundamental concept behind hydraulic systems, ensuring that pressure applied at one point is felt equally at another point through the fluid, allowing force to be transmitted and even multiplied.

Hydraulic Systems

Hydraulic systems operate based on the transmission of pressure through an incompressible fluid between pistons or interfaces of different sizes. By applying a small force over a smaller area, these systems can produce a larger force over a larger area, making them ideal for lifting and moving heavy loads.

Force, Pressure, and Area Relationship

The relationship between force, pressure, and area is described by the equation F = P × A, where F is the force exerted, P is the pressure applied, and A is the area over which the pressure is distributed. This fundamental equation is used in calculating the forces in hydraulic systems and understanding how pressure can generate significant lifting forces.

Gravitational Force Conversion

To relate the force produced by a hydraulic system to the mass it can support, it is necessary to convert the force (in newtons) to mass (in kilograms) by dividing by the acceleration due to gravity (approximately 9.8 m/s²). This conversion is essential in determining the maximum mass that a hydraulic lift can support based on the available lifting force.

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