Problem 16 - Applied Physics

Question

If a dentist's chair weighs \(16 \overline{0} 0 \mathrm{~N}\) and is raised by a large piston with crosssectional area \(75.0 \mathrm{~cm}^{2}\), what force must be exerted on a small piston of crosssectional area \(3.75 \mathrm{~cm}^{2}\) to lift the chair?

Step-by-Step Solution

Answer

The force needed on the small piston is 80 N.

Step 1: Understand the Problem

The problem requires us to find the force applied to a smaller piston to lift a chair supported by a larger piston. This relates to Pascal's principle, which states that pressure applied to a confined fluid is transmitted undiminished to all parts of the fluid and the walls of its container.

Step 2: Identify Known Quantities

We know the weight of the dentist's chair is the force exerted by the large piston, which is \(1600\,\mathrm{N}\). The area of the large piston is \(75.0\,\mathrm{cm}^2\) and the area of the small piston is \(3.75\,\mathrm{cm}^2\).

Step 3: Set Up the Equation Using Pascal's Principle

According to Pascal's principle, the pressure exerted by the pistons is equal: \[\frac{F_1}{A_1} = \frac{F_2}{A_2}\]where \(F_1\) is the force we need to find, \(F_2 = 1600\,\mathrm{N}\), \(A_1 = 3.75\,\mathrm{cm}^2\), and \(A_2 = 75.0\,\mathrm{cm}^2\).

Step 4: Solve for the Unknown Force

Rearrange the equation to solve for \(F_1\): \[F_1 = \frac{F_2 \cdot A_1}{A_2} \]Substitute the known values:\[F_1 = \frac{1600 \,\mathrm{N} \times 3.75\,\mathrm{cm}^2}{75.0\,\mathrm{cm}^2} = \frac{6000\,\mathrm{N}\cdot \mathrm{cm}^2}{75.0\,\mathrm{cm}^2} = 80\,\mathrm{N}\]

Step 5: Conclusion

The force that must be exerted on the small piston to lift the dentist's chair is \(80\,\mathrm{N}.\)

Key Concepts

hydraulic systems

Hydraulic systems are fascinating devices that use fluid to transmit force. These systems are often found in various forms of machinery, from construction equipment to brake systems in cars. They function based on Pascal's principle, which allows them to lift heavy objects with relatively little input force.

The core of a hydraulic system consists of two main components:

• A large piston, which is responsible for carrying the load.
• A small piston, which is where the input force is applied.

The pistons operate in cylinders filled with an incompressible fluid, usually oil. When a force is applied to the small piston, the pressure created in the fluid is transmitted to the large piston. This pressure can then lift a much greater force than what was originally applied, thanks to the difference in the surface area of the pistons.

For example, in the dentist's chair problem, the hydraulic system allows the chair's considerable weight to be lifted with little effort by applying force to a smaller piston.

pressure

Pressure is a measure of force applied on a surface per unit area, and it plays a crucial role in understanding hydraulic systems. In a hydraulic system, pressure is uniformly distributed throughout the fluid due to Pascal's principle.

The formula used to calculate pressure is:

\[ P = \frac{F}{A} \]

where:

• \( P \) is the pressure,
• \( F \) is the force applied,
• \( A \) is the area over which the force is applied.

In the context of hydraulic systems, when a force is applied to the small piston, the pressure created is transmitted throughout the fluid, reaching the larger piston. The beauty of this principle is that even though the force applied amounts to only 80 N in the aforementioned exercise, the large piston can carry the entire weight of the dentist's chair.

Therefore, mastering the concept of pressure is vital for efficiently understanding how hydraulic systems work, making it possible to manipulate the force necessary for lifting heavy loads efficiently.

force and area relationship

The relationship between force and area is a fundamental concept in physics, particularly essential in hydraulic systems. This relationship is clearly seen through Pascal's principle, which states that pressure in a fluid is constant. Thus, any increase in pressure at any point in a confined fluid is transmitted undiminished throughout the fluid.

In our dentist's chair example, we already know from the given solution that:
\[ \frac{F_1}{A_1} = \frac{F_2}{A_2} \]

where:

• \( F_1 \) is the force required on the small piston,
• \( A_1 \) is the area of the small piston,
• \( F_2 \) is the force exerted by the large piston, which is the weight of the chair (1600 N),
• \( A_2 \) is the area of the large piston.

This formula not only shows how forces relate to the areas of the pistons but also how manipulating these areas can amplify force. By increasing the area of the large piston, a smaller force on the smaller piston can lift a much heavier load. This interplay between force and area is what allows hydraulic systems to perform high-efficiency work with minimal input force.