Q89P - University Physics with Modern Physics

Question

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 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 $\mathbf{1}\mathbf{.}\mathbf{05}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}}\mathit{k}\mathit{g}\mathbf{/}{\mathit{m}}^{\mathbf{3}}$ . (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 required work done is $1.26\times {10}^{5}J$ .

b) The heart’s power output is $1.46W$ .

Step 1: Identify the given data

The volume of the blood is $V=7500L\left(\frac{{10}^{-3}{m}^3}{1L}\right)=7.500m^3$ .

The density of the blood is $\rho =1.05\times {10}^{3}kg/m^3$ .

Height of distance is $S=1.63m$ .

The time during the day is $t=1day\left(\frac{86400s}{1day}\right)=86400 s$ .

Step 1: Concept/Significance of Work done

The expression of work done is given by,

W = FS

W = mgS

W = pVgs…………………..(1)

Step 3: Determine the work done by heart in one day (a)

Substitute all the values in equation (1), and we get,

 

 $$\begin{gathered} W=\left(1.05\times {10}^{3}kg/m^3\right)\left(7.500m^3\right)\left(9.8m/s^2\right)\left(1.63\right) \\ =1.26\times {10}^{5}J \end{gathered}$$

 

Therefore, the required work done is $1.26\times {10}^{5}J$ .

Step 4: Determine the heart’s power output in watts (b)

The expression for power is given by,

$P=\frac{W}{t}$………………….(2)

 

Substitute all the values in equation (2), and we get,

 

$$\begin{gathered} P=\frac{1.26\times {10}^{5}J}{86400} \\ =1.46 W \end{gathered}$$ 

 

Therefore, the heart’s power output is 1.46 W .