Problem 18 - Applied Physics

Question

Compressed air in a car lift applies a force to a piston with radius \(5.00 \mathrm{~cm}\). This pressure is transmitted through a hydraulic system to a second piston with radius \(15.0 \mathrm{~cm}\). (a) How much force must the compressed air exert to lift a vehicle weighing \(1.33 \times 10^{4} \mathrm{~N}\) ? (b) What pressure produces the lift?

Step-by-Step Solution

Answer

\( F_1 = 1.48 \times 10^3 \text{ N} \), Pressure = 1.89 \times 10^5 \text{ Pa}.

Step 1: Understanding the problem

The problem involves a hydraulic system where we need to determine two things: the force exerted by the compressed air on the first piston to lift a vehicle, and the pressure required to produce this lift. We use the principle that pressure is transmitted equally in a closed hydraulic system.

Step 2: Using Pascal's Principle

According to Pascal's principle, in a hydraulic system, the pressure exerted on the first piston is transmitted equally to the second piston. Therefore, \( \text{Pressure on piston 1} = \text{Pressure on piston 2} \).

Step 3: Calculating the areas of the pistons

The area of each piston is given by the formula \( A = \pi r^2 \). For piston 1, \( r_1 = 5.00 \text{ cm} = 0.050 \text{ m} \), so \( A_1 = \pi (0.050)^2 \). For piston 2, \( r_2 = 15.0 \text{ cm} = 0.150 \text{ m} \), so \( A_2 = \pi (0.150)^2 \).

Step 4: Calculating force on the first piston

Using Pascal's principle, \( \frac{F_1}{A_1} = \frac{F_2}{A_2} \). Solving for \( F_1 \), we get \( F_1 = F_2 \times \frac{A_1}{A_2}\). Given \( F_2 = 1.33 \times 10^4 \text{ N} \), plug in the areas to find \( F_1 \).

Step 5: Calculating pressure

The pressure exerted by the compressed air on the first piston is \( P = \frac{F_1}{A_1} \). Use the value of \( F_1 \) found in step 4 and the area \( A_1 \) to compute this pressure.

Key Concepts

Pascal's Principle

Pascal's Principle is a fundamental concept used in hydraulic systems. It states that when pressure is applied to a confined fluid, this pressure is transmitted equally in all directions throughout the fluid. This principle makes it possible for hydraulic systems to amplify force.

In practice, this means that pressure applied to a small piston in a hydraulic lift is replicated on a larger piston, allowing even modest forces to lift heavy loads. For example, in the car lift system described in the problem, the pressure exerted by compressed air on a small piston is transferred to a larger piston lifting a vehicle. This effectively multiplies the input force without the need for large machinery.

Understanding Pascal's Principle is crucial because it explains why hydraulic systems are so efficient. It’s the reason why such systems can perform large, heavy load tasks with relatively little input force. This principle is essential whether you're inflating a tire or ensuring safe braking in vehicles.

Force Calculation

Calculating force in a hydraulic system involves understanding the relationship between force, area, and pressure, often derived from Pascal's Principle.

In the scenario of the exercise, we need to find out how much force the compressed air applies to the first piston to lift a heavy vehicle. Given the radius of the first piston is 5.00 cm, we first calculate the area of this piston using the formula:

• \( A_1 = \pi (r_1)^2 \)

where \( r_1 = 0.050 \text{ m} \).

Also, we know that the hydraulic system must lift a vehicle exerting a force of \( F_2 = 1.33 \times 10^4 \text{ N} \) on the second piston, which has a radius of 15.0 cm. The area \( A_2 \) is calculated similarly.

Using Pascal's Principle, which states that the pressure applied is the same throughout the system, we find the force on the first piston (\( F_1 \)) using:

• \( \frac{F_1}{A_1} = \frac{F_2}{A_2} \)

Thus:

• \( F_1 = F_2 \times \frac{A_1}{A_2} \)

By inserting the known values for the areas and the force needed to lift the vehicle, we find the amount of force the compressed air needs to apply.

Pressure Calculation

Pressure calculation in the hydraulic system is a step that ties everything together, ensuring the system functions correctly.

Once the force on the first piston (\( F_1 \)) is determined, we can calculate the pressure exerted by the compressed air. The pressure \( P \) is defined as the force applied per unit area, and it's calculated using the formula:

• \( P = \frac{F_1}{A_1} \)

where \( F_1 \) is the force exerted by the compressed air, and \( A_1 \) is the area of the first piston.

Calculating pressure helps understand how efficiently the system can lift the vehicle. It ensures that the input provided by the compressed air is adequate to generate the pressure needed to raise the car. This process underscores the principle of energy conservation in hydraulic systems.

By breaking down pressure calculation into easy steps, students can appreciate how practical applications relate back to foundational physics principles. This also reinforces the deeply interconnected nature of force, area, and pressure in engineering applications.