Problem 61 - University Calculus: Early Transcendentals

Question

The effect of flight maneuvers on the heart The amount of work done by the heart's main pumping chamber, the left ventricle, is given by the equation $$W=P V+\frac{V \delta v^{2}}{2 g}$$ where \(W\) is the work per unit time, \(P\) is the average blood pressure, \(V\) is the volume of blood pumped out during the unit of time, \(\delta\) ("delta") is the weight density of the blood, \(v\) is the average velocity of the exiting blood, and \(g\) is the acceleration of gravity. When \(P, V, \delta,\) and \(v\) remain constant, \(W\) becomes a function of \(g,\) and the equation takes the simplified form $$W=a+\frac{b}{g}(a, b \text { constant })$$. As a member of NASA's medical team, you want to know how sensitive \(W\) is to apparent changes in \(g\) caused by flight maneuvers, and this depends on the initial value of \(g\). As part of your investigation, you decide to compare the effect on \(W\) of a given change \(d g\) on the moon, where \(g=5.2 \mathrm{ft} / \mathrm{sec}^{2},\) with the effect the same change \(d g\) would have on Earth, where \(g=32 \mathrm{ft} / \mathrm{sec}^{2} .\) Use the simplified equation above to find the ratio of \(d W_{\text {moon }}\) to \(d W_{\text {Eurth }}\)

Step-by-Step Solution

Answer

The ratio of changes \( \frac{dW_{\text{moon}}}{dW_{\text{Earth}}} \) is approximately 37.82.

Step 1: Differentiate W with respect to g

Given the simplified equation for work as \( W = a + \frac{b}{g} \), differentiate \( W \) with respect to \( g \) to determine how changes in \( g \) affect \( W \). The differential of \( W \) with respect to \( g \) is \( \frac{dW}{dg} = -\frac{b}{g^2} \).

Step 2: Calculate the change in W on the moon

Using \( g = 5.2 \text{ ft/sec}^2 \) for the moon, substitute into the expression for \( \frac{dW}{dg} \). The rate of change of work on the moon is \( \frac{dW_{\text{moon}}}{dg} = -\frac{b}{(5.2)^2} \).

Step 3: Evaluate the change in W on Earth

Using \( g = 32 \text{ ft/sec}^2 \) for Earth, substitute into the expression for \( \frac{dW}{dg} \). The rate of change of work on Earth is \( \frac{dW_{\text{Earth}}}{dg} = -\frac{b}{32^2} \).

Step 4: Find the ratio of the changes in W

To find the ratio \( \frac{dW_{\text{moon}}}{dW_{\text{Earth}}} \), divide the rate of change on the moon by the rate of change on Earth: \( \frac{dW_{\text{moon}}}{dW_{\text{Earth}}} = \frac{-\frac{b}{(5.2)^2}}{-\frac{b}{32^2}} = \frac{(32)^2}{(5.2)^2} \).

Step 5: Simplify the ratio expression

Calculate \( \left(\frac{32}{5.2}\right)^2 \) to find the ratio of changes in \( W \) due to a change in \( g \); this equals \( \left(\frac{32}{5.2}\right)^2 \approx 37.82 \).

Key Concepts

Differentiation

Differentiation is all about measuring how a certain quantity changes in response to a change in another variable. In our problem, we are interested in how the work done by the heart changes with gravitational acceleration, denoted as \( g \). The given function for work \( W \) is represented as \( W = a + \frac{b}{g} \). To differentiate \( W \) with respect to \( g \), we apply the power rule to obtain the derivative \( \frac{dW}{dg} = -\frac{b}{g^2} \).
This tells us that as the acceleration due to gravity \( g \) increases, the rate of work done by the heart decreases. The differentiation operation helps us understand the sensitivity of work relative to changes in gravity. It's crucial for calculating how the heart responds under different gravitational forces.

Work

In physics, 'work' is a term that describes the amount of energy transferred by a force when it moves an object. When it comes to the heart, the left ventricle does work to pump blood throughout the body. The formula \( W = PV + \frac{V \delta v^{2}}{2 g} \) represents the work per unit time done by the ventricle. It comprises two parts:

• \( PV \): This term represents the work done against blood pressure \( P \) to eject blood volume \( V \).
• \( \frac{V \delta v^{2}}{2 g} \): This part accounts for the kinetic energy aspect, relating to blood velocity \( v \), density \( \delta \), and the gravitational field \( g \).

The simplified form \( W = a + \frac{b}{g} \) shows how work depends on gravity when all other factors remain constant. This simplification allows us to focus specifically on gravity's impact, crucial for understanding scenarios like those encountered in flight maneuvers.

Gravity

Gravity is a fundamental force that attracts two bodies towards each other, commonly observed as the force holding us to the Earth's surface. It plays a pivotal role in our heart's pumping function. Our exercise examines how variations in gravitational acceleration \( g \) - like those experienced during flight - affect the heart's work rate.
The original work equation \( W = a + \frac{b}{g} \) integrates gravity into its calculations. On Earth, gravity is approximately \( 32 \text{ ft/sec}^2 \), while on the moon, it's about \( 5.2 \text{ ft/sec}^2 \). Variations in gravity lead to changes in the heart's workload since \( W \) inversely depends on \( g \). This means, with lower gravity, such as on the moon, the heart has to exert less effort compared to on Earth.

Blood Pressure

Blood pressure is an essential factor influencing how the heart works. It represents the force exerted by circulating blood upon the walls of blood vessels. In the context of our problem, the parameter \( P \) in the work equation \( W = PV + \frac{V \delta v^{2}}{2 g} \) stands for average blood pressure, influencing the heart's workload directly.
Keeping \( P \) constant allows us to isolate and study the effect of changes in gravitational pull on the heart’s work by focusing on \( W = a + \frac{b}{g} \). Understanding how these variables interplay is vital for assessing how the heart adapts to different stressors, like varying gravity during intense maneuvers, without the added complexities of fluctuating blood pressure.