Question

The lower end of a long plastic straw is immersed below the surface of the water in a plastic cup. An average person sucking on the upper end of the straw can pull water into the straw to a vertical height of 1.1 m above the surface of the water in the cup. (a) What is the lowest gauge pressure that the average person can achieve inside his lungs? (b) Explain why your answer in part (a) is negative.

Short answer

(a) -10791 Pa; (b) Negative because suction lowers pressure below atmospheric.

Step-by-step solution

Understand the Problem

We have a straw that can pull water to a height of 1.1 meters. We need to find the lowest gauge pressure that allows this to happen. We will use fluid pressure concepts to solve this.

Recall Concepts of Pressure

Gauge pressure measures the difference from atmospheric pressure. In this problem, the gauge pressure will be the pressure needed to support the water column in the straw.

Apply Fluid Pressure Formula

The formula for pressure due to a column of liquid is given by:\[ P = \rho g h \]where \( \rho \) is the density of water (1000 kg/m³), \( g \) is the acceleration due to gravity (9.81 m/s²), and \( h \) is the height of the water column (1.1 m).

Calculate Pressure

Substitute the known values:\[ P = (1000 \text{ kg/m}^3)(9.81 \text{ m/s}^2)(1.1 \text{ m}) \]Calculate to find:\[ P = 10791 \text{ Pa} \]

Determine Gauge Pressure

Gauge pressure is defined as the pressure relative to atmospheric pressure, which is approximately 101325 Pa at sea level. To obtain gauge pressure, we consider the pressure supporting the column relative to atmospheric pressure:\[ P_{gauge} = P - P_{atm} = -10791 \text{ Pa} \]

Explain Why Gauge Pressure is Negative

The gauge pressure is negative because the pressure inside the straw must be lower than atmospheric pressure to lift the water. This negative gauge pressure corresponds to the suction needed to raise the water.

Key concepts

Gauge Pressure

Gauge pressure is a concept used to describe the pressure of a fluid that is measured relative to the atmospheric pressure around it. This is crucial because most pressure gauges measure this relative pressure under the assumption that atmospheric pressure is present. In the context of our exercise, gauge pressure represents the difference between the pressure inside the lung and the surrounding atmospheric pressure.

The key takeaway about gauge pressure is that it can be negative, as observed when a person sucks water up a straw. The pressure inside the lungs needs to be lower than the atmospheric pressure to draw water up. Thus, when a person creates a pressure lower than the surrounding area, it results in a negative gauge pressure. This is why when we calculate the gauge pressure for this scenario, it comes out to be -10791 Pa.

In short, gauge pressure helps us understand and measure how much more or less pressure there is compared to the surrounding environment, giving us valuable insight into fluid dynamics.

Atmospheric Pressure

Atmospheric pressure is the pressure exerted by the weight of the atmosphere above us. It acts as a baseline for other pressure measurements, including gauge pressure. At sea level, atmospheric pressure is approximately 101325 Pa. It's this baseline that allows us to understand how much the pressure within a system, like a straw, differs from the surrounding environment.

In scenarios involving fluid dynamics, such as our straw example, atmospheric pressure plays a critical role. When you suck water through the straw, you are effectively reducing the pressure inside the straw. This reduced pressure compared to the constant atmospheric pressure allows the water to rise. Atmospheric pressure is what pushes on the liquid's surface outside the straw, enabling the movement of water based on the created pressure difference.

Understanding atmospheric pressure is crucial because it allows us to use simpler calculations for gauge pressure and helps assess the actual movement of fluids in various contexts.

Fluid Pressure Formula

The fluid pressure formula is central to solving problems like the one presented in the exercise. The formula is described as follows: \[ P = \rho g h \]where:

• \( \rho \) is the fluid density, for water it is about 1000 kg/m³.
• \( g \) is the gravitational pull on the fluid, approximately 9.81 m/s².
• \( h \) is the height of the fluid column, as seen in our example, 1.1 m.

This formula helps us in determining the pressure exerted by a fluid column when it's in a vertical position, such as in a straw.

By applying this equation, one can calculate the absolute pressure that a fluid exerts at the bottom of that column. In the context of the exercise, substituting the known values into the formula gives us the pressure required to maintain a column of water standing at 1.1 m tall. It's essential to grasp this formula, as it simplifies the process of determining pressures in various fluid scenarios.

Learning to use this formula effectively aids in understanding key fluid dynamics concepts, crucial for many real-world applications.