Question

Pressure and the doctor's office In this problem we examine the hydrostatic pressure in a fluid at rest. (a) Consider a small fluid element \(\Delta x \times \Delta y \times \Delta z\) at rest. Write down the balance of forces on the fluid element due to the fluid pressure \(p\) and the gravitational pull of the Earth. Show that that this leads to the differential equation $$\nabla p=\rho \mathbf{g}$$ where \(g\) is the acceleration due to gravity. (b) Solve the differential equation derived in (a) assuming a uniform fluid density. Show that the pressure in the fluid is given by \(p(z)-p_{0}-\rho g z,\) where the z-axis is in the direction opposite of \(\mathbf{g}\) (c) Estimate the atmospheric pressure, (Hint: Look up the density of air and make a reasonable guess for the helght of the atmosphere.) (d) If you raise your arm above your head as the doctor is measuring your blood pressure, how much will the measurement change compared with when you keep your arm level with the heart. Try it next time you are at the doctor's officel

Short answer

The pressure change when you raise your arm from heart level is explained by the hydrostatic pressure difference, which is determined by the density of the blood, acceleration due to gravity, and the difference in height.

Step-by-step solution

Deriving the differential equation

Consider a small fluid element \(\Delta x \times \Delta y \times \Delta z\) that is experiencing a pressure difference across its height \(\Delta z\). The pressure at the top and bottom faces of this fluid parcel differ by approximately \(\Delta p = (\nabla p) \Delta z\), where \(\nabla p\) is the gradient of the pressure. The gravitational force on the element is given by \(\Delta F = \rho g \Delta V\), where \(\Delta V\) is the volume of the fluid parcel, \( \Delta x \times \Delta y \times \Delta z = \Delta V\). Balancing these forces, we find that \(\Delta F=\Delta p \Delta A\), where \(\Delta A\) is the cross-sectional area of the element. Dividing through by \(\Delta z\), we obtain the hydrostatic balance equation, \(\nabla p = \rho g\).

Solving the differential equation

Now we solve the differential equation \(\nabla p = \rho g\). Integrate both sides with respect to \(z\). On the left side, we have \(\int dp = \int \rho g dz\). Evaluating these integrals gives \(p(z) - p_0 = -\rho g z\), where \(p_0\) is the pressure at \(z = 0\). Solving for \(p(z)\), we find the pressure at any height in the fluid: \(p(z) = p_0 + \rho g z\).

Estimating the atmospheric pressure

To estimate atmospheric pressure, consider the atmosphere to be a fluid of air with uniform density. Look up the density of air \(ρ_{air}\) about 1.225 kg/m³, and take \(g = 9.81m/s²\) and the typical height of the atmosphere to be about 10,000 m, we can estimate the pressure at the surface of the earth as \(p_0 + \rho_{air} g h\).

Effect on blood pressure measurement

The blood pressure measurement would change if the arm is raised above the head compared to at the heart level due to change in hydrostatic pressure. The change in pressure would be \(\Delta P = \rho g \Delta z\), where \(\rho\) is the density of the blood, \(g\) is the acceleration due to gravity, and \(\Delta z\) is the difference in height between the two points (the length of your arm). Try it next time at the doctor's office!

Key concepts

Fluid Mechanics

Fluid mechanics is a branch of physics that deals with the behavior and properties of fluids, either in motion or at rest. It encompasses both liquids and gases. The study of fluid mechanics is essential for understanding various phenomena in engineering and natural sciences. In this context, hydrostatic pressure becomes a key concept.

Hydrostatics, a subfield of fluid mechanics, considers fluids at rest. The foundational principle here is that changes in pressure in a static fluid result in a balance of forces. This directly leads to important principles like Pascal's Law, which states that pressure applied to a confined fluid is transmitted undiminished in every direction.

When handling exercises in fluid mechanics, as evident in the given problem, one often deals with elements like pressure differentials, gravitational pull, and equilibrium of forces. Understanding fluid mechanics, gets simpler as we divide fluids into small elements and analyze these individually, thus helping establish basic laws and equations that can be applied universally.

Differential Equation

A differential equation is a mathematical equation that involves derivatives of a function. Such equations are vital in describing physical phenomena because they relate various rates of change to different variables.

In the hydrostatic pressure problem, a differential equation is derived by analyzing the forces acting on a small fluid element in equilibrium. The balance between the pressure difference across this element and the gravitational force it experiences results in the equation: \( abla p = \rho \mathbf{g} \). This equation states that the gradient of pressure equals the product of fluid density and gravitational acceleration.

Solving such differential equations, as shown in the step-by-step solution, involves integrating both sides. This process enables us to calculate how the pressure changes with height in a fluid. For example, integrating the given equation leads to the formula: \( p(z) = p_0 + \rho g z \), which describes how pressure varies with depth or height, an essential aspect for calculating fluid pressures in various conditions.

Gravitational Force

Gravitational force is a fundamental force of nature that attracts two bodies with mass towards each other. It's responsible for various phenomena ranging from the orbit of planets to the fall of an apple from a tree. In fluid mechanics, gravity plays a crucial role in determining the behavior of fluid elements.

In the exercise, gravitational force is considered when analyzing the forces acting on a small fluid slice. The downward gravitational pull is calculated using \( \Delta F = \rho g \Delta V \), where \( \rho \) represents fluid density, and \( \Delta V \) is the volume of the fluid element. This force is the reason why pressures vary with depth, i.e., the pressure in a fluid increases as one moves downward.

The concept further helps in creating a relationship between pressure and height, represented in the differential equation \( abla p = \rho g \). This equation signifies that gravitational force is directly responsible for the pressure differences experienced in a fluid at rest.

Atmospheric Pressure

Atmospheric pressure is the pressure exerted by the weight of air in the Earth's atmosphere. It affects weather patterns and is crucial for various biological and mechanical processes. The standard atmospheric pressure at sea level is approximately 101,325 Pascals.

Estimating atmospheric pressure involves considering the atmosphere as a continuous fluid with distributed weight over any object on the Earth's surface. Using the previously discussed differential equation, you can calculate pressure at a point due to atmospheric air. Knowledge of the density of air (approximately 1.225 kg/m³) and standard gravity (9.81 m/s²) allows you to make accurate estimates.

In the exercise, calculating atmospheric pressure substantially contributes to understanding how outside pressures influence measurements and conditions seen in medical and engineering settings.

Blood Pressure Measurement

Blood pressure measurement is a critical medical procedure used to determine the pressure exerted by circulating blood on the walls of blood vessels. When measuring blood pressure, understanding how hydrostatic pressure affects readings is crucial.

When you raise your arm above the level of the heart, as questioned in the exercise, hydrostatic pressure changes. Blood pressure cuffs typically read more when positioned at a lower height, due to the added pressure exerted by the column of fluid (blood) between the cuff and the heart.

This change can be estimated using \( \Delta P = \rho g \Delta z \), where \( \Delta z \) represents the difference in height. It's a perfect demonstration of how fundamental fluid mechanics applies to everyday medical procedures, offering practical insights into how posture can affect diagnostics and the importance of consistent positioning during medical assessments.