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At a given instant, the blood pressure in the heart is \(1.6 \times 10^{4} \mathrm{Pa}\) If an artery in the brain is 0.45 m above the heart, what is the pressure in the artery? Ignore any pressure changes due to blood flow.

Short Answer

Expert verified
The pressure in the artery is approximately \(1.14 \times 10^{4} \mathrm{Pa}\).

Step by step solution

01

Understand the Concept

The pressure in a fluid decreases with height due to gravity. This concept can be described by the hydrostatic pressure equation, which is used to relate pressures at different heights in a fluid column.
02

Identify Given Values

The initial blood pressure at the heart is given as \(P_0 = 1.6 \times 10^{4} \mathrm{Pa}\). The height difference \(h\) between the heart and the artery is 0.45 m.
03

Apply the Hydrostatic Pressure Equation

Use the hydrostatic pressure equation: \(P = P_0 - \rho gh\), where \(P\) is the pressure at the height \(h\), \(\rho\) is the density of the blood (assuming \(1050 \; \mathrm{kg/m^3}\)), \(g\) is the acceleration due to gravity \(9.81 \; \mathrm{m/s^2}\), and \(h\) is the height difference.
04

Substitute the Values

Insert the known values into the hydrostatic pressure equation: \[P = 1.6 \times 10^{4} - (1050)(9.81)(0.45)\] This calculates the pressure in the artery.
05

Calculate the Result

Perform the calculation: \[P = 1.6 \times 10^{4} - 4634.25\] \[P = 1.136 \times 10^{4} \; \mathrm{Pa}\] Thus, the pressure in the artery is approximately \(1.14 \times 10^{4} \mathrm{Pa}\) (rounded to three significant figures).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Blood Pressure
Blood pressure is the force that blood exerts on the walls of blood vessels. It is a vital indicator of the condition of the circulatory system. Understanding blood pressure involves two main measurements: systolic and diastolic pressure. Systolic pressure is the highest pressure in your arteries when your heart beats, while diastolic pressure is the lowest pressure when your heart rests between beats. In medical practice, blood pressure is often measured in millimeters of mercury (mmHg), but in fluid mechanics and physics, it's common to see it represented in pascals (Pa). A healthy blood pressure reading is considered to be around 120/80 mmHg. In this exercise, they used pascals to calculate the difference in pressure due to the height of the blood column. Just as in the heart, the blood pressure at different locations in the body can change based on distance from the pumping source, which is a primary concern in understanding blood flow to various parts of the body, like the brain.
Fluid Mechanics
Fluid mechanics is the branch of physics dealing with the behavior of fluids (liquids and gases) and the forces on them. It provides fundamental principles to understand and describe the flow and pressure within any fluid, including blood as it flows through the circulatory system. In fluid mechanics, one significant concept is the idea of hydrostatic pressure. This pressure is the force exerted by a fluid at equilibrium due to its weight, explained by the hydrostatic pressure equation. The equation considers the fluid's density, gravitational acceleration, and the height of the fluid column. Understanding these principles ensures better grasping of how pressure phenomena occur not just in physics problems but also in everyday life, such as how blood pressure changes from the heart to the brain. These principles apply directly to understanding how the heart pumps blood throughout the body, how pressure changes, and how these changes can affect overall health.
Pressure Calculation
Calculating pressure in fluid systems like blood flow involves specific formulas that take into account fluid characteristics and environmental conditions. The hydrostatic pressure equation \[ P = P_0 - \rho gh \] is a standard tool used to determine the pressure at different points in a fluid column.
  • \( P_0 \): Original pressure at the source (often the heart in medical contexts)
  • \( \rho \): Density of the fluid, such as blood (approximately \(1050 \, \text{kg/m}^3\))
  • \( g \): Gravitational acceleration (\(9.81 \, \text{m/s}^2\))
  • \( h \): Height difference between the source and point of interest
Using these variables, you can find the pressure difference due to gravity's effect on the fluid column. For example, in the given exercise, values were substituted to find the blood pressure in a brain artery relative to the heart. By understanding how to calculate such pressures, students and professionals can predict and analyze various fluid systems, such as water towers, pipelines, and even human circulatory systems.

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