Problem 12 - Applied Physics

Question

The small circular piston of a hydraulic press has an area of \(8.00 \mathrm{~cm}^{2} .\) If the applied force is \(25.0 \mathrm{~N}\), what must the area of the large piston be to exert a pressing force of \(3650 \mathrm{~N}\) ?

Step-by-Step Solution

Answer

The area of the large piston must be \(1168 \mathrm{~cm}^2\).

Step 1: Understanding the Basic Principle

In a hydraulic system, pressures in identical liquids are transmitted equally. The fundamental principle is expressed by the equation: \( \frac{F_1}{A_1} = \frac{F_2}{A_2} \), where \( F_1 \) and \( F_2 \) are forces, while \( A_1 \) and \( A_2 \) are the areas of the small and large pistons, respectively.

Step 2: Substitute Known Values

We are given:- \( F_1 = 25.0 \mathrm{~N} \)- \( A_1 = 8.00 \mathrm{~cm}^{2} \)- \( F_2 = 3650 \mathrm{~N} \)Substitute these values into the equation: \( \frac{25.0}{8.00} = \frac{3650}{A_2} \).

Step 3: Solve for the Large Piston Area

Rearrange the equation to solve for \( A_2 \):\( A_2 = \frac{3650 \times 8.00}{25.0} \).

Step 4: Calculate the Area

Perform the multiplication and division:\( A_2 = \frac{29200}{25.0} \).Calculate the result to get the value of \( A_2 \).

Step 5: Determine the Final Answer

After calculation, we find \( A_2 = 1168 \mathrm{~cm}^{2} \). This is the area of the large piston necessary to exert a pressing force of \( 3650 \mathrm{~N} \).

Key Concepts

Pressure Transmission

In hydraulic systems, the concept of pressure transmission is essential. Imagine you have liquid inside a sealed container. When you apply pressure to this liquid, it gets transmitted throughout without any reduction.
This is because liquids are incompressible under normal conditions. So, the pressure at one point in the liquid is exactly equal at all other points. In its simplest terms, this is how hydraulic systems magnify force.
The equation used to analyze this concept is \( \frac{F_1}{A_1} = \frac{F_2}{A_2} \).

• \( F_1 \) is the force applied to the small piston.
• \( A_1 \) is the area of the small piston.
• \( F_2 \) is the force exerted by the large piston.
• \( A_2 \) is the area of the large piston.

When you push down on the small piston, it creates pressure which then gets transferred. This means the larger the area, the greater the force output, provided the pressure remains constant.
Understanding this principle allows for designs that efficiently move or lift heavy objects even with relatively small input forces.

Force and Area Relationship

In hydraulic systems, the force applied is directly related to the area of the piston through which it is applied, according to pressure equations.
By rearranging the equation \( F = P \times A \), where \( P \) is pressure, we can see how force is proportionate to area.
This relationship is vital because it provides a clear understanding of how much force might be expected given the piston size:

• If the area increases, the force exerted increases for the same pressure.
• Conversely, less area requires more pressure to achieve the same force.

The Implication in Hydraulic Design

When considering the design of hydraulic machinery, such as brakes or lifts, knowing how force translates into area is fundamental.
Engineers use this principle to calculate the necessary dimensions of pistons to ensure that desired forces are applied without failure.
Therefore, the precision in these calculations is crucial, making sure structures are both efficient and safe.

Pascal's Principle

Pascal's Principle forms the basis of hydraulic systems. This principle states that any pressure applied to a confined fluid is transmitted undiminished through the fluid in all directions. This means if you apply pressure at any point in a closed liquid system, that pressure is relayed equally throughout every part of the system.
This universal transmission of pressure is what allows hydraulic systems to work.
With a smaller input force on a smaller area, a much larger force can be exerted by a larger area, demonstrating significant mechanical advantage.

Applications and Importance

When using devices like a car lift or hydraulic press, Pascal's Principle ensures the applied force is multiplied, allowing for efficient performance of heavy-duty tasks without needing immense input forces.
This principle is also why brakes in vehicles can exert tremendous stopping power with just a small push on the brake pedal. By understanding and applying Pascal's Principle, a wide range of machinery and equipment can operate effectively and safely.
Measurement precision in hydraulic designs benefits applications from automotive technologies to large-scale industrial machinery, showcasing its versatile utility.