Problem 89 - University Physics with Modern Physics

Question

The human heart is a powerful and extremely reliable pump. Each day it takes in and discharges about 7500 L of blood. Assume that the work done by the heart is equal to the work required to lift this amount of blood a height equal to that of the average American woman (1.63 m). The density (mass per unit volume) of blood is \(1.05 \times 10^3 \, \mathrm{kg/m}^3\). (a) How much work does the heart do in a day? (b) What is the heart's power output in watts?

Step-by-Step Solution

Answer

(a) The heart does about 1.262 × 10^5 J of work daily. (b) Its power output is approximately 1.459 W.

Step 1: Calculate the Mass of the Blood Pumped

The density of blood is given by \( 1.05 \times 10^3 \, \mathrm{kg/m}^3 \). The volume of blood pumped per day is 7500 liters, which is equivalent to \( 7500 \, \mathrm{L} = 7.5 \, \mathrm{m}^3 \). To find the mass, use the formula: \( \text{mass} = \text{density} \times \text{volume} \). Thus, \( \text{mass} = 1.05 \times 10^3 \, \mathrm{kg/m}^3 \times 7.5 \, \mathrm{m}^3 = 7875 \, \mathrm{kg} \).

Step 2: Calculate the Work Done by the Heart

Work done is given by \( \text{work} = \text{force} \times \text{distance} \). The force exerted here is the weight of the blood, which is \( \text{mass} \times \text{gravity} \). With gravity \( g = 9.8 \, \mathrm{m/s}^2 \) and the lift height being 1.63 meters: \( \text{work} = 7875 \, \mathrm{kg} \times 9.8 \, \mathrm{m/s}^2 \times 1.63 \, \mathrm{m} = 1.262 \times 10^5 \, \mathrm{J} \).

Step 3: Calculate the Power Output of the Heart

Power is defined as the rate of doing work. Given that the heart works over a 24-hour period, convert this time to seconds: \( 24 \times 3600 = 86400 \) seconds. The power \( P \) can be calculated as \( P = \frac{\text{work}}{\text{time}} \). Thus, \( P = \frac{1.262 \times 10^5 \, \mathrm{J}}{86400 \, \mathrm{s}} \approx 1.459 \) watts.

Key Concepts

Understanding Heart Function in Energy Terms

The human heart is often compared to an engine, constantly working to pump blood throughout the body. Each day, it moves roughly 7500 liters of blood, tirelessly circulating it through our vascular system. In our exercise, we consider the work done by the heart as equivalent to the energy required to lift this volume of blood to a height of 1.63 meters, which represents the average height of a woman in the U.S.

In physics, work is calculated as force times distance. Here, the force is the weight of the blood. Weight, a type of force, is the mass of an object multiplied by gravitational acceleration. This means that despite the heart's seemingly effortless function, it's doing a remarkable amount of mechanical work continuously, similar to lifting weights repeatedly. This perspective helps us appreciate the sheer energy expended by this vital organ every single day.

Calculating Power Output

Power is a term we often hear in various contexts, but in physics, it refers to the rate at which work is done or energy is transferred. For the heart, its power output is determined by how much energy it can exert over a set period of time. In our exercise, after calculating the total work done by the heart in joules (which is about 126,200 joules), we determine how much of this work is done every second to find the heart's power output in watts.

Since the heart operates continuously non-stop, this process involves converting the work done over a whole day into work done per second. To do this, we recognize a day comprises 86,400 seconds. Dividing the total energy by time yields the heart's power output, which is approximately 1.459 watts. This might seem small compared to mechanical devices, but considering the heart works autonomously and consistently over decades without rest, this power output underscores its incredible capability.

Density and Mass Conversion Insight

When dealing with fluids such as blood, density is a crucial concept. Density is the mass per unit volume and is expressed in units such as kilograms per cubic meter (kg/m³). In our scenario, the given density of blood is 1050 kg/m³, a typical value for human blood.

To determine the mass of the blood pumped by the heart in a day, we first convert the given volume in liters to cubic meters since the density is in terms of cubic meters. With a sheer volume of 7500 liters or 7.5 cubic meters of blood being pumped daily, the mass of the blood calculates to 7875 kg using the density formula. This conversion is significant as it links volume to mass and is essential in expanding our understanding of how bodily functions translate into physical work. Understanding and performing these conversions are key whenever calculating other similar biological or ecological scenarios.