Problem 75 - College Physics

Question

\(\bullet\) \(\bullet\) The power of the human heart. 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 that amount of blood a height equal to that of the average Ameri- can female, approximately 1.63 \(\mathrm{m} .\) The density of blood is 1050 \(\mathrm{kg} / \mathrm{m}^{3} .\) (a) How much work does the heart do in a day? (b) What is the heart's power output in watts? (c) In fact, the heart puts out more power than you found in part (b). Why? What other forms of energy does it give the blood?

Step-by-Step Solution

Answer

The heart does approximately 125,955.4 J of work per day with a power output of 1.46 W. Other forms of energy include kinetic energy and pressure maintenance.

Step 1: Convert Blood Volume to Mass

First, convert the daily volume of blood (7500 L) to cubic meters (m³) using the fact that 1 L = 0.001 m³. Then, calculate the mass of the blood using the density: \[ \text{Volume} = 7500 \, \text{L} \times 0.001 \, \frac{\text{m}^3}{\text{L}} = 7.5 \, \text{m}^3 \] \[ \text{Mass} = \text{Density} \times \text{Volume} = 1050 \, \frac{\text{kg}}{\text{m}^3} \times 7.5 \, \text{m}^3 = 7875 \, \text{kg} \]

Step 2: Calculate the Work Done by the Heart

The work done to lift the blood can be calculated using the formula for gravitational potential energy: \[ \text{Work} = \text{Mass} \times \text{Gravity} \times \text{Height} \] where the gravitational acceleration \( g \) is approximately 9.81 m/s²:\[ \text{Work} = 7875 \, \text{kg} \times 9.81 \, \frac{\text{m}}{\text{s}^2} \times 1.63 \, \text{m} \] \[ \text{Work} \approx 125,955.4 \, \text{J} \]

Step 3: Calculate the Heart's Power Output

Power is defined as the work done per unit time. Calculate the power output over a day (24 hours): \[ \text{Power} = \frac{\text{Work}}{\text{Time}} = \frac{125,955.4 \, \text{J}}{86400 \, \text{s}} \approx 1.46 \, \text{W} \]

Step 4: Consider Other Forms of Energy

The heart's actual power output is greater than calculated, due to additional energies involved. Besides raising blood, the heart also imparts kinetic energy to blood flow and overcomes vascular resistance. Thus, energy is used in moving blood through arteries and veins.

Key Concepts

Gravitational Potential Energy

Gravitational potential energy is the energy possessed by an object due to its position relative to the Earth. For the human heart, this concept helps us understand how lifting blood against gravity all day is a form of work the heart must do.

The formula for gravitational potential energy is given by:

\[ E_p = mgh \]

where:

• \(E_p\) is the gravitational potential energy.
• \(m\) is the mass of the object (in this case, the blood).
• \(g\) is the acceleration due to gravity (approximately 9.81 m/s²).
• \(h\) is the height to which the object is lifted.

In our scenario, the heart pumps around 7,875 kg of blood daily to a height of 1.63 meters. This is akin to lifting the blood to a level that's a bit over 5 feet, equivalent to the average American female's height. The energy used in this process can be substantial when calculated over an entire day.

Blood Density

Blood density plays a vital role in calculating the mass of the blood being pumped. It's the measure of how compact or concentrated the blood is.

The given blood density is 1050 kg/m³. This value allows us to convert the volume of blood the heart handles in a day into mass, which is crucial for further calculations in terms of work and power. Using the conversion of volume from liters to cubic meters (7.5 m³ for 7500 liters), we determine the mass by multiplying the blood's density with this volume, resulting in a mass of 7,875 kg.

Through this, we see how blood density directly influences the heart's workload because greater density means the heart has to exert more force to move the same blood volume.

Energy Conversion

Energy conversion is the process of changing one form of energy into another. In the context of the heart, it involves converting chemical energy from the body's metabolism into mechanical energy to pump blood.

While calculating the work done, we realize the heart not only lifts the blood, as seen in the gravitational potential energy calculations, but also does so by converting different energy forms. The heart, as a biological engine, constantly transforms the energy we derive from food (chemical energy) into the force and movement required to circulate blood (mechanical energy).

This is pivotal in understanding the complex mechanisms by which our body maintains homeostasis, ensuring every part receives necessary oxygen and nutrients efficiently.

Physiology and Mechanics

The heart's physiology and mechanics are crucial to understanding its function as a pump. As one of the most vital organs, the heart pumps blood continuously, demonstrating incredible efficiency and endurance.

From a mechanical perspective, the heart's design allows it to handle various forms of resistance like arterial pressure and blood viscosity. Its muscle fibers contract rhythmically, enabling it to perform the repetitive task of pushing blood against gravity and through vast networks of blood vessels.

Besides gravitational work, the heart also imparts kinetic energy to the blood, ensuring smooth and efficient circulation. This kinetic energy contributes significantly to overall power output, alongside overcoming vascular resistance. Thus, the true power of the heart goes beyond simple lifting tasks, encapsulating a range of mechanical and physiological processes involved in maintaining life.