Question

An airtight box having a lid with an area of \(12 \mathrm{in} .{ }^{2}\) is partially evacuated. If a force of \(108 \mathrm{lb}\) is required to pull the lid off the box, and the outside atmospheric pressure is \(15 \mathrm{lb} / \mathrm{in}^{2}\), what was the pressure in the box?

Short answer

The pressure inside the box is \(6 \mathrm{lb/in}^{2}\)

Step-by-step solution

Understanding the problem

The definition of pressure is force divided by area. We know the area of the box's lid, the outside pressure, and the force necessary to pull off the lid. We can use this information to find the pressure inside the box.

Calculate pressure due to force

The pressure, \(P_f\), due to the force required to lift the box lid can be calculated using the formula for pressure as \(P_f=F/A\), where F is the force and A is the area. We can substitute the known values \(F = 108 \mathrm{lb}\) and \(A = 12 \mathrm{in}^{2}\) to get \(P_f = 108 \mathrm{lb} / 12 \mathrm{in}^{2} = 9 \mathrm{lb/in}^{2}\)

Calculate the pressure inside the box

The total pressure on the lid is a combination of the outside pressure and the pressure within the box due to the force required to lift the lid. Therefore, by simple addition, the pressure inside the box will be the total outside pressure minus the pressure caused by the force. If the total outside pressure \(P_o\) is \(15 \mathrm{lb/in}^{2}\) and the pressure due to the force \(P_f\) is \(9 \mathrm{lb/in}^{2}\), then the pressure inside the box, \(P_i\) can be calculated as \(P_i = P_o - P_f = 15 \mathrm{lb/in}^{2} - 9 \mathrm{lb/in}^{2} = 6 \mathrm{lb/in}^{2}\)

Key concepts

Pressure Calculation

Understanding the calculation of pressure is fundamental in physics and engineering. It is defined as the force applied per unit area and is represented by the equation:
Pressure (P) = Force (F) / Area (A).

In the exercise, we apply this concept to determine the pressure inside an airtight box. By knowing the force that is required to lift the lid and the area over which this force acts, we can calculate the pressure exerted by the force, which is a crucial step in solving the problem. It's important to use consistent units when calculating pressure to get a meaningful result.

The beauty of this calculation lies in its simplicity and its ability to be applied in a variety of situations, from determining the pressure inside a sealed container, as in the exercise, to calculating the force needed to hold a hydraulic press steady.

Force and Pressure Relationship

The relationship between force and pressure is direct and straightforward. When you apply a force over a specific area, you exert pressure. This relationship is governed by the formula:

P = F / A.

In other words, pressure is directly proportional to the force applied and inversely proportional to the area over which it is spread. This means that if you apply the same force over a smaller area, you will increase the pressure. Understanding this relationship helps to explain the step-by-step solution in our exercise, where we figured out the pressure in the box by dividing the force needed to remove the lid by the area of the lid. It’s an invaluable concept for students in various scientific disciplines, including physics, engineering, and even medicine.

Atmospheric Pressure

Atmospheric pressure is the force exerted by the weight of air in the Earth's atmosphere on a surface. It's important in our exercise as it affects the total pressure acting on the outside of the box lid.

Atmospheric pressure at sea level is commonly estimated to be about 14.7 pounds per square inch (psi), but it can vary depending on altitude and weather conditions. In the problem, the atmospheric pressure was given as 15 psi, which played a critical role in calculating the pressure difference between the inside and the outside of the box and ultimately in determining the pressure inside the evacuated box.

Understanding atmospheric pressure is not just relevant for homework problems but also for everyday phenomena, like why a straw works when you drink a beverage or why your ears pop when you ascend in an airplane.

Pressure Units

Measurements of pressure are expressed in various units, depending on the context and geographical location. In the United States, pressure is often measured in pounds per square inch (psi), as used in our exercise. Other common units include the pascal (Pa), the standard unit of pressure in the International System of Units (SI).

One pascal is defined as one newton per square meter. For higher pressures, it’s common to see kilopascals (kPa) or even megapascals (MPa). Atmospheric pressure, for example, is often referred to as 101.325 kPa.

Understanding different units of pressure and how to convert between them is vital in science and engineering practices, ensuring clear communication and accurate computation across different systems of measurement.