Problem 11 - Fundamentals of Physics

Question

\(\cdot 11\) As Giruffe bending to drink. In a giraffe with its head \(2.0 \mathrm{~m}\) above its heart, and its heart \(2.0 \mathrm{~m}\) above its feet, the (hydrostatic) gauge pressure in the blood at its heart is 250 torr. Assume that the giraffe stands upright and the blood density is \(1.06 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\). In torr (or \(\mathrm{mm} \mathrm{Hg}\) ), find the (gauge) blood pressure (a) at the brain (the pressure is enough to perfuse the brain with blood, to keep the giraffe from fainting) and (b) at the feet (the pressure must be countered by tight-fitting skin acting like a pressure stocking). (c) If the giraffe were to lower its head to drink from a pond without splaying its legs and moving slowly, what would be the increase in the blood pressure in the brain? (Such action would probably be lethal.)

Step-by-Step Solution

Answer

(a) 232 torr, (b) 268 torr, (c) 18 torr increase in brain pressure.

Step 1: Understanding the problem

To find the blood pressure at different points in the giraffe, we use the hydrostatic pressure formula. The gauge pressure is given for the giraffe's heart, and we want to calculate it at the brain and feet. The formula for hydrostatic pressure is \( p = p_0 + \rho gh \), where \( p_0 \) is the initial pressure at the heart, \( \rho \) is the density of the blood, \( g \) is the acceleration due to gravity, and \( h \) is the height difference.

Step 2: Convert the pressure units

The gauge pressure at the heart is given as 250 torr. First, we need to convert this to Pascals (Pa) to use in calculations. Knowing that 1 torr = 133.322 Pa, the pressure at the heart in Pascals is \( 250 \times 133.322 \).

Step 3: Calculate pressure at the brain

The brain is 2 meters above the heart. Use the hydrostatic pressure formula to find the pressure drop: \( \Delta p = \rho g h = 1.06 \times 10^3 \times 9.81 \times 2 \). Subtract this from the heart pressure to find the pressure at the brain. Convert the result back to torr by dividing by 133.322.

Step 4: Calculate pressure at the feet

The feet are 2 meters below the heart. Again, use the hydrostatic pressure formula to compute the pressure increase: \( \Delta p = \rho g h = 1.06 \times 10^3 \times 9.81 \times 2 \). Add this to the heart pressure to find the pressure at the feet. Convert the result back to torr as previously.

Step 5: Calculate pressure increase when drinking

When the giraffe lowers its head to drink, the head is another 2 meters below its heart. Calculate the additional pressure using \( \Delta p = \rho g h = 1.06 \times 10^3 \times 9.81 \times 2 \), and convert this increase into torr.

Key Concepts

Gauge Pressure

Gauge pressure is an essential concept in understanding fluids and pressures in various contexts, like blood flow in a giraffe. Unlike absolute pressure, which measures against a perfect vacuum, gauge pressure measures pressure relative to the surrounding atmospheric pressure. This makes it particularly useful in scenarios where only the difference in pressure matters, such as inside the body.
In the giraffe exercise, gauge pressure at the heart is used as the reference point (250 torr or mm Hg). When calculating the gauge pressure relative to different parts of the giraffe’s body, such as the brain or the feet, we account for the changes in blood pressure due to height.
Hydrostatic pressure, which relies on the fluid density, gravity, and height (\( \Delta p = \rho g h \)), is a crucial factor in determining these differences in pressure. In this way, we only need to measure how pressure changes from one point to another rather than comparing it to an absolute baseline.

Blood Density

Blood density plays a vital role in calculating hydrostatic pressure changes in the body. In the exercise, the blood density of the giraffe is given as \(1.06 \times 10^3 \, \text{kg/m}^3\). This density shows how much mass is present within a specific volume of blood, which directly impacts how pressure changes with height.
The denser the blood, the higher the pressure difference will be due to gravity. As the blood moves upward toward the brain in the giraffe, the pressure reduces because the gravitational component acts downward. Similarly, when moving downward toward the feet, the pressure increases due to the accumulating weight of blood.
Understanding blood density is crucial for applications in physiology and medicine, ensuring that the pressure at critical points in the body, like the brain, is sufficient for proper blood flow, as well as in engineering to design appropriate devices or garments, such as the fittings mimicking pressure stockings described in the exercise.

Pressure Conversion

Pressure conversion is an important skill in physics and engineering, ensuring that we can accurately work across different units of measurement. In most calculations, like the giraffe's blood pressure example, pressure starts in torr but needs to be calculated in Pascals (Pa) using the conversion: 1 torr = 133.322 Pa.
This conversion allows for accurate calculations using the hydrostatic pressure formula, which typically requires standard SI units (Pascals).
Once the hydrostatic pressure difference is calculated in Pascals, results often need to be converted back to the original units (torr) for a more straightforward interpretation in contexts like medicine where mm Hg is standard.
Through practice, mastering unit conversion contributes greatly to understanding and precision, particularly in interdisciplinary fields like biomedical engineering, enhancing the ability to design solutions such as medical equipment that needs precise pressure calculations.