Problem 9 - Fundamentals of Physics

Question

Blood pressure in Argentinosaurus. (a) If this longnecked, gigantic sauropod had a head height of \(21 \mathrm{~m}\) and a heart height of \(9.0 \mathrm{~m}\), what (hydrostatic) gauge pressure in its blood was required at the heart such that the blood pressure at the brain was 80 torr (just enough to perfuse the brain with blood)? Assume the blood had a density of \(1.06 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\). (b) What was the blood pressure (in torr or \(\mathrm{mm} \mathrm{Hg}\) ) at the feet?

Step-by-Step Solution

Answer

(a) 1013.18 torr, (b) 1714.62 torr.

Step 1: Understand the Problem

We have to find the gauge pressure at the heart needed to maintain 80 torr at the brain, given the height difference of 12 m (since 21 m - 9 m = 12 m), and the pressure at the feet given the height difference of 9m (due to both positions).

Step 2: Calculate Pressure Drop Due to Height Difference

We use the hydrostatic pressure formula: \( \Delta P = \rho g h \), where \( \rho = 1.06 \times 10^3 \, \text{kg/m}^3 \), \( g = 9.8 \, \text{m/s}^2 \), and \( h = 12 \text{ m} \). So, \( \Delta P = 1.06 \times 10^3 \, \text{kg/m}^3 \times 9.8 \, \text{m/s}^2 \times 12 \, \text{m} \).

Step 3: Simplify the Pressure Calculation

Calculate \( \Delta P = 1.24416 \times 10^5 \, \text{Pa} \). Since \( 1 \text{ torr} = 133.322 \text{ Pa} \), convert \( \Delta P \) from pascals to torrs: \( \Delta P \approx \frac{1.24416 \times 10^5 \, \text{Pa}}{133.322 \, \text{Pa/torr}} \approx 933.18 \, \text{torr} \).

Step 4: Find Required Heart Pressure

To maintain 80 torr at the brain, add this pressure to the pressure drop: \( P_{heart} = 80 \, \text{torr} + 933.18 \, \text{torr} \approx 1013.18 \, \text{torr} \). This is the gauge pressure needed at the heart.

Step 5: Calculate Pressure at the Feet

For the pressure at the feet, using the heart-to-feet height difference, calculate: \( \Delta P = 1.06 \times 10^3 \, \text{kg/m}^3 \times 9.8 \, \text{m/s}^2 \times 9 \, \text{m} = 93528 \, \text{Pa} \). Convert this: \( \Delta P \approx \frac{93528 \, \text{Pa}}{133.322 \, \text{Pa/torr}} \approx 701.44 \, \text{torr} \), and add it to the heart pressure: \( P_{feet} = 1013.18 \, \text{torr} + 701.44 \, \text{torr} = 1714.62 \, \text{torr} \).

Step 6: Summary of Results

The gauge pressure required at the heart is approximately 1013.18 torr to just perfuse the brain. The blood pressure at the feet is approximately 1714.62 torr.

Key Concepts

Blood Pressure in Dinosaurs

Imagine if you were a gigantic Argentinosaurus, towering over 21 meters, keeping your blood pumping would be quite a task! This massive dinosaur needed a significant blood pressure to ensure its brain received enough blood.
The concept of hydrostatic pressure plays a crucial role here. Due to the height difference between the heart and the brain, the dinosaur's heart had to exert extra effort to push blood against gravity.
This is known as hydrostatic pressure, calculated using the formula \(\Delta P = \rho g h\), where:

• \(\rho\) is the blood density \(1.06 \times 10^3 \, \text{kg/m}^3\).
• \(g\) represents the gravity \(9.8 \, \text{m/s}^2\).
• \(h\) stands for the height difference from heart to brain (12 meters in this case).

By calculating the pressure difference, scientists estimated the heart's necessary pressure to ensure enough blood flow to the brain, around 1013.18 torr. This showcases the incredible cardiovascular adaptation needed by such gigantic creatures to sustain their enormous size.

Gauge Pressure

Gauge pressure is one of the essential measurements used in understanding how pressure works within a system, such as a dinosaur's cardiovascular structure. Unlike absolute pressure, gauge pressure is the pressure relative to the atmospheric pressure around it.
When considering the Argentinosaurus, gauge pressure is critical to understand how much additional pressure the dinosaur's heart needed to exert. This pressure ensures that blood reaches the brain at an adequate level required for perfusion.
The atmospheric pressure at sea level is typically 760 torr. The gauge pressure needed at the heart, calculated as 1013.18 torr, indicates how much the internal pressure exceeds this atmospheric base. This difference helps scientists and biologists comprehend how efficiently this ancient creature's cardiovascular system operated, countering gravitational forces to deliver oxygen-rich blood to its brain.

Pressure Conversion

Converting pressure measurements between different units is a critical skill in physics and biology, especially when studying systems like the blood circulation in dinosaurs. For the Argentinosaurus, we dealt with pressures in both pascals and torrs, requiring conversion.
Pressure can be measured in various units:

• Pascals (Pa), the SI unit for pressure.
• Torr, a unit named after Evangelista Torricelli, related to the atmospheric pressure.

To convert the pressure calculation of 1.24416 \( \times \) \(10^5 \, \text{Pa}\) into torrs, we used the conversion factor of 1 torr = 133.322 Pa. Thus, 933.18 torr was the pressure in terms of the height difference from heart to brain.
Understanding how to switch between these units helps us to accurately express and analyze pressures experienced by biological entities, keeping the head and body properly perfused with blood in different environments.