Problem 52 - Physics

Question

At reviews the concept that plays the central role in this problem. (a) The volume flow rate in an artery supplying the brain is \(3.6 \times 10^{-6} \mathrm{~m}^{3} / \mathrm{s}\). If the radius of the artery is \(5.2 \mathrm{~mm}\), determine the average blood speed. (b) Find the average blood speed at a constriction in the artery if the constriction reduces the radius by a factor of \(3 .\) Assume that the volume flow rate is the same as that in part (a).

Step-by-Step Solution

Answer

The average blood speed in the artery is 0.042 m/s, and at the constriction, it is 0.383 m/s.

Step 1: Understanding the Problem

We need to find the average blood speed in an artery given the volume flow rate and the radius. Then, we will find the average blood speed after the artery constricts and its radius decreases by a factor of 3. The volume flow rate remains the same in both scenarios.

Step 1: Calculate Cross-Sectional Area of Artery

The cross-sectional area of a cylinder (such as an artery) is given by the formula for the area of a circle: \( A = \pi r^2 \). Given that the radius of the artery is \(5.2 \, \text{mm} = 0.0052 \, \text{m}\), the area is:\[ A = \pi (0.0052)^2 \approx 8.5 \times 10^{-5} \, \text{m}^2 \]

Step 2: Calculate Average Blood Speed in Normal Artery

The average blood speed (\( v \)) can be calculated using the formula for volume flow rate (\( Q \)): \( Q = A \cdot v \). Solving for \( v \), we get:\[ v = \frac{Q}{A} = \frac{3.6 \times 10^{-6} \, \text{m}^3/\text{s}}{8.5 \times 10^{-5} \, \text{m}^2} \approx 0.042 \, \text{m/s} \]

Step 3: Calculate New Radius at Constriction

The radius at the constriction is reduced by a factor of 3, so the new radius is:\[ r_{\text{new}} = \frac{0.0052}{3} \, \text{m} \approx 0.00173 \, \text{m} \]

Step 4: Calculate Cross-Sectional Area at Constriction

Using the reduced radius, the new cross-sectional area is:\[ A_{\text{new}} = \pi (0.00173)^2 \approx 9.4 \times 10^{-6} \, \text{m}^2 \]

Step 5: Calculate Average Blood Speed at Constriction

Using the same volume flow rate, the average blood speed at the constriction is:\[ v_{\text{new}} = \frac{Q}{A_{\text{new}}} = \frac{3.6 \times 10^{-6} \, \text{m}^3/\text{s}}{9.4 \times 10^{-6} \, \text{m}^2} \approx 0.383 \, \text{m/s} \]

Key Concepts

Volume Flow Rate

Volume flow rate is a measure of the volume of fluid passing through a section of a conduit, like an artery, per unit time. In this context, we're looking at blood flow in an artery, which brings essential nutrients and oxygen to various parts of the body. Volume flow rate is often represented by the symbol \( Q \). For our problem, the flow rate in the artery is given as \( 3.6 \times 10^{-6} \, \text{m}^3/\text{s} \).

• This indicates that in one second, \( 3.6 \times 10^{-6} \) cubic meters of blood pass through the artery.
• The constant flow rate over varying artery sizes helps demonstrate how other variables must adjust to maintain consistent blood transport.

Understanding volume flow rate is critical since it influences how effectively blood can travel through varying artery sizes without causing issues like turbulence or blockages.

Cross-Sectional Area

The cross-sectional area of an artery refers to the area of the slice through the artery. Imagine cutting through an artery to view its circular shape — this circular view is the cross-section. Its area is calculated using the formula for the area of a circle, \( A = \pi r^2 \). For our problem, the initial artery radius is \( 5.2 \, \text{mm} = 0.0052 \, \text{m} \).

• Applying this radius in the formula gives a cross-sectional area \( A \approx 8.5 \times 10^{-5} \, \text{m}^2 \).
• The cross-sectional area is crucial because it determines how much space blood has to pass through at any given moment.

As we saw later in the problem when the artery has a constriction, the radius decreased, hence reducing the cross-sectional area. This relationship is central to understanding how blood flow dynamics change with artery constriction.

Average Blood Speed

Average blood speed refers to how fast the blood moves through an artery and depends on both the volume flow rate and the cross-sectional area. The relationship between the three is defined by the equation \( v = \frac{Q}{A} \), where \( v \) is the average blood speed, \( Q \) is the volume flow rate, and \( A \) is the cross-sectional area.

• In a normal artery with \( Q = 3.6 \times 10^{-6} \, \text{m}^3/\text{s} \) and \( A \approx 8.5 \times 10^{-5} \, \text{m}^2 \), the average blood speed \( v \approx 0.042 \, \text{m/s} \).
• Knowing the average blood speed can help in diagnosing or predicting conditions in circulatory health.

Faster blood speeds occur when the cross-sectional area decreases, as observed during artery constriction, while maintaining the same flow rate.

Artery Constriction

Artery constriction refers to the narrowing of the artery, typically due to physiological or pathological changes. In this exercise, the problem models a scenario where the artery radius is reduced by a factor of 3, impacting the cross-sectional area and consequently the blood flow dynamics.

• The new radius becomes \( 0.00173 \, \text{m} \), and hence the new cross-sectional area is approximately \( 9.4 \times 10^{-6} \, \text{m}^2 \).
• This reduction in area results in an increase in the average blood speed to \( 0.383 \, \text{m/s} \).
• This is an important illustration of how the circulatory system adapts to deliver required volumes of blood despite physical restrictions.

Whether due to natural physiological adaptation or medical conditions, understanding artery constriction aids in assessing cardiovascular health and response to stressors.