Problem 22 - Fundamentals of Physics

Question

In dogfights. When a pilot takes a tight turn at high speed in a modern fighter airplane, the blood pressure at the brain level decreases, blood no longer perfuses the brain, and the blood in the brain drains. If the heart maintains the (hydrostatic) gauge pressure in the aorta at 120 torr (or \(\mathrm{mm} \mathrm{Hg}\) ) when the pilot undergoes a horizontal centripetal acceleration of \(4 g\), what is the blood pressure (in torr) at the brain, \(30 \mathrm{~cm}\) radially inward from the heart? The perfusion in the brain is small enough that the vision switches to black and white and narrows to "tunnel vision" and the pilot can undergo \(g\) - \(\operatorname{LOC}(" g\) induced loss of consciousness"). Blood density is \(1.06 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\).

Step-by-Step Solution

Answer

The blood pressure at the brain is 26.4 torr.

Step 1: Understand the Problem

We need to determine the blood pressure at the brain when a pilot experiences a horizontal centripetal acceleration of \(4g\). The heart maintains a pressure of 120 torr at the aorta. The distance from the heart to the brain is 30 cm radially inward. Given the blood density, we must calculate the pressure difference due to this centripetal force and hydrostatic pressure.

Step 2: Identify Known Values

We have the following known values:- Gauge pressure at the aorta: 120 torr- Acceleration: \(4g = 4 \times 9.81 \text{ m/s}^2\)- Distance from heart to brain: 0.30 m- Blood density: \(\rho = 1.06 \times 10^3 \text{ kg/m}^3\)

Step 3: Calculate Pressure Difference Due to Centripetal Acceleration

The pressure difference between the aorta and the brain can be calculated using the equation for hydrostatic pressure due to an acceleration:\[\Delta P = \rho \cdot a \cdot h\]where \(\Delta P\) is the pressure difference, \(\rho\) is the density of blood, \(a\) is the acceleration \(4g = 4 \times 9.81 \text{ m/s}^2\), and \(h\) is the distance (0.30 m).

Step 4: Convert Acceleration to Pressure Difference

First, calculate the acceleration in m/s^2: - \(a = 4 \times 9.81 = 39.24 \text{ m/s}^2\)Now calculate \(\Delta P\):\[\Delta P = 1.06 \times 10^3 \text{ kg/m}^3 \times 39.24 \text{ m/s}^2 \times 0.30 \text{ m} = 1.24 \times 10^4 \text{ Pa}\]

Step 5: Convert Pressure to Torr

1 torr is equivalent to 133.322 Pa. We convert the pressure difference calculated above to torr:\[\Delta P = \frac{1.248 \times 10^4}{133.322} \approx 93.6 \text{ torr}\]

Step 6: Calculate Blood Pressure at the Brain

The pressure at the brain is the pressure at the aorta minus the pressure difference:\[\text{Pressure at brain} = 120 \text{ torr} - 93.6 \text{ torr} = 26.4 \text{ torr}\]

Step 7: Conclusion

The blood pressure at the brain, when subjected to a \(4g\) horizontal acceleration, drops significantly to 26.4 torr, explaining why a pilot might experience tunnel vision or black and white vision under such conditions.

Key Concepts

Centripetal acceleration

In physics, centripetal acceleration is a crucial concept when examining objects moving in a circular path. It describes the inward acceleration that keeps an object moving along this path.
Centripetal acceleration ensures that rather than traveling in a straight line due to inertia, the object continues to veer inwards, maintaining its circular motion. The magnitude of centripetal acceleration can be calculated by the formula: \[ a_c = \frac{v^2}{r} \]where:

• \(a_c\) is the centripetal acceleration,
• \(v\) is the velocity of the object,
• \(r\) is the radius of the circular path.

When a pilot makes a tight turn, as explained in the exercise, the effect of centripetal acceleration becomes crucial. At higher speeds, the required centripetal force increases, causing a considerable stress on the person operating the vehicle. In such scenarios, the change in blood pressure in various parts of the body is significant, emphasizing the need to understand the physical forces at play.

Blood pressure calculation

Understanding and calculating blood pressure within the human body is essential, especially when dealing with unusual situations like high gravitational (g-force) environments. Blood pressure is fundamentally the force exerted by circulating blood on the walls of blood vessels. In problem-solving, especially in physics, we often deal with gauge pressure, which measures pressure relative to atmospheric pressure.
In this exercise, the heart maintains a certain pressure at the aorta. To find the pressure at the brain, we subtract the pressure difference caused by centripetal acceleration from aorta pressure. The key calculation involves this formula: \[ \Delta P = \rho \cdot a \cdot h \]where:

• \(\Delta P\) is the change in pressure between two points,
• \(\rho\) is the density of blood,
• \(a\) is the acceleration (often due to gravity),
• \(h\) is the height or distance between the two points.

It's crucial to convert our units appropriately, like converting pressure from Pascals to torr, for consistency in results. This understanding helps detect critical situations, such as where blood pressure drops enough to risk consciousness loss, as seen in high-speed scenarios.

Hydrostatic pressure

Hydrostatic pressure is a concept in fluid dynamics that measures the pressure exerted by a fluid at equilibrium due to gravity or other accelerative forces. Hydrostatic pressure becomes an integral part of physics and engineering, particularly in calculating fluid behavior in varying gravitational fields.
In simple terms, it is the pressure exerted by a fluid on a submerged object or at a specific depth within the fluid. The formula often used to calculate this pressure in a liquid at rest is:\[P = \rho \cdot g \cdot h\]where:

• \(P\) is the hydrostatic pressure,
• \(\rho\) is the fluid density,
• \(g\) is the acceleration due to gravity,
• \(h\) is the depth or height of the fluid column.

In the exercise, hydrostatic principles apply both due to distance (height) and additional forces such as centripetal acceleration. Additionally, when considering the blood pressure in the pilot's scenario, hydrostatic principles help illuminate why there is a significant pressure variation from the heart to the brain during high-speed maneuvers.