The heart of a man pumps $5 ext{ L}$ of blood through the arteries per minute at a pressure of $150 ext{ m

Question

The heart of a man pumps $5 	ext{ L}$ of blood through the arteries per minute at a pressure of $150 	ext{ mm}$ of mercury. If the density of mercury is $13.6 	imes 10^3 	ext{ kg/m}^3$ and $g = 10 	ext{ m/s}^2$, then the power of heart in watt is:

Accepted Answer

1. **Identify the Formula:** The power ($P_{ower}$) delivered by a pump (like the heart) moving a fluid is given by the product of the pressure ($P$) and the volume flow rate ($\frac{dV}{dt}$).
   $$ P_{ower} = P 	imes \frac{dV}{dt} $$
   (Derived from $P_{ower} = \vec{F} \cdot \vec{v} = (P \cdot A) \cdot v = P \cdot (Av) = P \cdot \frac{dV}{dt}$) 
2. **Convert Units to SI:**
   - Volume flow rate: $\frac{dV}{dt} = 5 	ext{ L/min} = \frac{5 	imes 10^{-3} 	ext{ m}^3}{60 	ext{ s}} = \frac{5}{6} 	imes 10^{-4} 	ext{ m}^3/	ext{s}$.
   - Pressure head ($h$): $150 	ext{ mm} = 0.15 	ext{ m}$.
   - Density ($ho$): $13.6 	imes 10^3 	ext{ kg/m}^3$.
   - Gravity ($g$): $10 	ext{ m/s}^2$.
3. **Calculate Pressure ($P$):** Using the hydrostatic pressure formula:
   $$ P = hho g = 0.15 	imes (13.6 	imes 10^3) 	imes 10 $$
   $$ P = 20400 	ext{ N/m}^2 	ext{ (Pa)} $$
   
4. **Calculate Power:**
   $$ P_{ower} = 20400 	imes \left( \frac{5 	imes 10^{-3}}{60} ight) $$
   $$ P_{ower} = \frac{20400 	imes 5}{60000} = \frac{102}{60} = 1.70 	ext{ W} $$

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