Problem 95 - University Physics with Modern Physics

Question

BIO Power of the Human Heart. The human heart is a powerful and extremely reliable pump. Each day it takes in and discharges about 7500 \(\mathrm{L}\) 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 \((1.63 \mathrm{m}) .\) The density (mass per unit volume) of blood is \(1.05 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}\) . (a) How much work does the heart do in a day? (b) What is the heart's power output in watts?

Step-by-Step Solution

Answer

The heart does approximately 1.26 x 10^5 J of work per day with a power output of about 1.46 W.

Step 1: Calculate the Volume in Cubic Meters

First, convert the volume of blood from liters to cubic meters. Since there are 1000 liters in a cubic meter, the blood volume in cubic meters is:\[ V = \frac{7500 \, \text{L}}{1000 \, \text{L/m}^3} = 7.5 \, \text{m}^3 \]

Step 2: Calculate the Mass of the Blood

Use the density formula, \( \text{mass} = \text{density} \times \text{volume} \), to find the mass of the blood as follows:\[ m = 1.05 \times 10^3 \, \text{kg/m}^3 \times 7.5 \, \text{m}^3 = 7875 \, \text{kg} \]

Step 3: Calculate the Work Done by the Heart

The work done, \( W \), on the blood is the gravitational potential energy, given by \( W = mgh \), where:- \( m = 7875 \, \text{kg} \) (mass of the blood)- \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity)- \( h = 1.63 \, \text{m} \) (height to lift the blood)Calculate:\[ W = 7875 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 1.63 \, \text{m} = 1.26 \times 10^5 \, \text{J} \]

Step 4: Determine the Power Output of the Heart

Power, \( P \), is the work done per unit time. Convert the daily work into a rate by determining the number of seconds in a day (24 hours, 60 minutes per hour, 60 seconds per minute):\[ t = 24 \times 60 \times 60 = 86400 \, \text{s} \] The power output is then calculated as:\[ P = \frac{1.26 \times 10^5 \, \text{J}}{86400 \, \text{s}} \approx 1.46 \, \text{W} \]

Key Concepts

Work and Energy

In the context of the human heart, understanding work and energy is crucial to grasp how its incredible daily tasks are achieved. Work, in physics, is defined as the force acting on an object times the distance over which the force is applied. For the heart, this means the work is equivalent to the energy needed to move blood against the force of gravity.
Every time the heart beats, it effectively does work by pushing blood up to a certain height inside the body.
This work is directly calculated by the formula: \( W = mgh \)

• \( W \) = Work done in joules (J)
• \( m \) = Mass of the blood in kilograms (kg)
• \( g \) = Acceleration due to gravity \( 9.8 \, \text{m/s}^2 \)
• \( h \) = Height in meters that the blood is lifted

For the heart to pump blood to a height equivalent to the size of an average human, it requires considerable energy, illustrated as work done against gravitational force. This way, it keeps blood circulating throughout the body, demonstrating a fascinating interplay of work and energy in physiology.

Power Calculation

The power output of the heart is another essential aspect of its functionality. Power in physics refers to the rate of doing work or the work done per unit of time.
The formula to calculate power is: \( P = \frac{W}{t} \)

• \( P \) = Power in watts (W)
• \( W \) = Work done in joules (J)
• \( t \) = Time in seconds (s)

To understand the heart's power, we need to see how much work it does over an entire day. Once we know the work done, we divide it by the number of seconds in a day to determine the power.
Through such calculations, it's revealed that the heart operates at a power level sufficient to keep blood moving efficiently, roughly equivalent to a low-wattage light bulb, yet critical to life. This insight highlights the heart's remarkable efficiency and relentless operation.

Density of Blood

The concept of density is integral to calculations involving the mass of blood that the heart pumps. Density is defined as mass per unit volume, given by the formula: \( \text{Density} = \frac{\text{mass}}{\text{volume}} \) Density allows us to convert between volumes of blood and the mass needing to be moved.
For blood, a typical density is \( 1.05 \times 10^{3} \text{kg/m}^3 \).
Given a certain volume of blood, by applying the density, we can calculate how much mass the heart has to work against gravity to move daily.

• This conversion is critical because it quantifies the actual load the heart must move.
• Knowing this mass helps in accurately assessing the work and energy required for circulation.

Understanding density thus enhances our comprehension of both blood flow and the heart's capabilities in maintaining blood circulation.

Gravitational Potential Energy

Gravitational potential energy is energy stored by objects due to their position relative to a gravitational field. For the heart, it is the energy required to lift blood to a certain height against gravity.
The formula used is: \( E_{gravitational} = mgh \)

• \( E_{gravitational} \) = Gravitational potential energy (Joules)
• \( m \) = Mass of the blood (kg)
• \( g \) = Acceleration due to gravity \( 9.8 \, \text{m/s}^2 \)
• \( h \) = Height lifted (m)

In the context of the heart, each lift of the blood to the body's height converts into gravitational potential energy.
By consistently overcoming gravity, the heart ensures that energy is effectively used to keep blood circulating.
This process demonstrates how the heart doesn't just pump blood, but also manages energy efficiently to sustain life-essential functions.