Problem 24 - College Physics

Question

Blood pressure. Systemic blood pressure is expressed as the ratio of the systolic pressure (when the heart first ejects blood into the arteries) to the diastolic pressure (when the heart is relaxed): systemic blood pressure \(=\frac{\text { systolic pressure }}{\text { diastolic pressure }}\) Both pressures are measured at the level of the heart and are expressed in millimeters of mercury (or torr), although the units are not written. Normal systemic blood pressure is \(\frac{120}{80}\) . (a) What are the maximum and minimum forces (in newtons) that the blood exerts against each square centimeter of the heart for a person with normal blood pressure? (b) As pointed out in the text, blood pressure is normally measured on the upper arm at the same height as the heart. Due to therapy for an injury, a patient's upper arm is extended 30.0 \(\mathrm{cm}\) above his heart. In that position, what should be his systemic blood pressure reading, expressed in the standard way, if he has normal blood pressure? The density of blood is 1060 \(\mathrm{kg} / \mathrm{m}^{3}\) .

Step-by-Step Solution

Answer

(a) Maximum force = 1.6 N, Minimum force = 1.067 N. (b) Adjusted blood pressure reading is 118/78 mmHg.

Step 1: Convert Pressure to Force

To find the force exerted by the blood, we convert the pressure in mmHg to Pascals (SI unit of pressure). We use the conversion factor: 1 mmHg = 133.322 Pa. For the maximum force: systolic pressure = 120 mmHg. Thus, 120 mmHg = 120 x 133.322 Pa. Then, convert to force using the formula: Force = Pressure x Area. Here, Area = 1 cm² = 0.0001 m². Therefore, maximum force = 120 x 133.322 x 0.0001 N.

Step 2: Calculate Maximum Force

Using the previous step, calculate the maximum force: \(120 \times 133.322 \times 0.0001 = 1.6 \text{ N}\).

Step 3: Calculate Minimum Force

Now, calculate the minimum force using the same method for diastolic pressure (80 mmHg): Convert 80 mmHg to Pascals: \(80 \times 133.322 \text{ Pa}\). Then calculate the force: \(80 \times 133.322 \times 0.0001 = 1.067 \text{ N}\).

Step 4: Understand the Effect of Height

Use the blood density and gravity to calculate the change in pressure due to height: Pressure change = density x gravity x height. Here, density = 1060 kg/m³, gravity = 9.81 m/s², height = 0.3 m (30 cm). Plug in the values to calculate the pressure difference at 30 cm above the heart.

Step 5: Calculate Pressure Difference

Pressure difference due to height: \(\Delta P = 1060 \times 9.81 \times 0.3 = 311.598 \text{ Pa} = \frac{311.598}{133.322} \text{ mmHg}\approx 2.34 \text{ mmHg}\).

Step 6: Adjust Blood Pressure Reading

To adjust the reading for a position 30 cm above the heart, subtract the pressure difference from both systolic and diastolic pressures. Normal pressures are 120/80 mmHg, so adjusted pressures are: \(120 - 2.34 \text{ and } 80 - 2.34\text{ mmHg}\). Resulting in a blood pressure reading of approximately 117.66/77.66 mmHg.

Key Concepts

Systolic Pressure

Systolic pressure refers to the peak pressure in the arteries when the heart's ventricles contract. It represents the moment when the heart actively pumps blood into the arteries, supplying oxygen-rich blood to the entire body. This is typically the first, higher number in a blood pressure reading. A normal systolic pressure is approximately 120 mmHg.

During this phase, the arterial walls must endure a surge of force as blood flows through. Understanding systolic pressure is crucial, as consistently high measurements can indicate potential cardiovascular issues such as hypertension. Keeping an eye on this number helps in the early detection and management of heart-related conditions.

It's important to maintain a balanced lifestyle, including proper diet and regular exercise, to support healthy systolic pressure levels.

Diastolic Pressure

Diastolic pressure is the lower number in a blood pressure report, indicating the pressure in the arteries while the heart rests between beats. This period is when the heart fills with blood and prepares for the next contraction. A normal diastolic pressure rests around 80 mmHg.

Understanding diastolic pressure is vital as it helps in assessing how well the heart relaxes and fills with blood. Consistently high diastolic measurements may suggest that the arteries are stiff, leading to increased cardiovascular risk. Monitoring both systolic and diastolic pressures provides a comprehensive view of cardiovascular health.

Making lifestyle changes, such as managing stress and avoiding excessive sodium intake, can help maintain a healthy diastolic pressure.

Pressure Conversion

Blood pressure is conventionally measured in millimeters of mercury (mmHg), a unit stemming from older barometer scales. However, in scientific analyses and calculations, converting these units to Pascals (Pa) — the SI unit of pressure — is often necessary for more accurate force assessments.

The conversion factor between mmHg and Pa is crucial: 1 mmHg equals 133.322 Pa. For instance, converting a systolic pressure of 120 mmHg into Pascals involves multiplying by this factor, equating to 15998.64 Pa. Practical applications of this conversion process allow for precise calculations in physics and engineering contexts, where SI units are standard.

Understanding how to perform pressure conversions is important, whether dealing with medical equipment calibrations or studying cardiovascular dynamics.

Force Calculation

Force calculation in the context of blood pressure involves determining the force exerted by blood on the walls of arteries. The formula used is: \[ \text{Force} = \text{Pressure} \times \text{Area} \]

This calculation requires the pressure in Pascals and the area in square meters. For example, if the systolic pressure is 120 mmHg, converting this to Pascals gives 15998.64 Pa. Assuming the artery area is 1 cm² (or 0.0001 m²), the force exerted is calculated as 1.6 N.

Understanding how to calculate force from pressure allows for better analysis of cardiovascular dynamics, aiding in research and medical diagnostics. This knowledge helps comprehend how various factors, like artery size or plaque buildup, impact blood movement and heart workload.

Effects of Height on Pressure

Height differences in the body can impact blood pressure due to gravitational forces. When measuring blood pressure at a height different than the heart, adjustments must be made. This is because the blood pressure can increase or decrease depending on whether the site of measurement is above or below the heart.

The formula to calculate this pressure change is:\[ \Delta P = \text{density} \times \text{gravity} \times \text{height} \] For a 30 cm elevation above the heart, using a blood density of 1060 kg/m³ and gravity (9.81 m/s²), the pressure change amounts to 311.598 Pa, or approximately 2.34 mmHg.

Understanding these effects is crucial for accurate blood pressure readings and necessary when assessing pressures in any setting where height varies, such as in different body positions or altitudes.