Question

The human heart discharges $75c{{m}^{3}}$ of blood per beat against an average pressure of 10cm of Hg. Assuming that the pulse frequency is 75 per minute, the power of the heart is
A. 1.25 W
A. 12.5 W
C. 0.125 W
D. 125 W

Answer 1

Hint:Power of a body is equal to the product of force on the body and the velocity of the body. Use the formula for force, velocity, volume and obtain an expression for power. Then calculate the power of the heart by substituting the given values.

Formula used:
$Power=Fv$
where F is force and v is velocity.
$F=PA$
where P is pressure and A is the area on which the pressure is applied.
$v=\dfrac{d}{t}$
where d is the distance travelled for time t.
$V=Ad$
where V is volume.

Complete step by step answer:
Power of a body is equal to $Power=Fv$ …. (i)
We know that force on a body can be written as $F=PA$.
Substitute the value of F in equation (i).
Then this means that $Power=Fv=PAv$ ….. (ii)
We also know that speed is equal to distance moved in one unit of time.
i.e. $v=\dfrac{d}{t}$
Substitute the value of v in equation (ii).
$Power=PA\dfrac{d}{t}$ …. (iii)
But we also know that the product of area and distance is equal to the volume covered by the body.
i.e. $V=Ad$
Now, substitute the value of V in equation (iii).
$Power=\dfrac{PV}{t}$.
It is given that the heart beats for 75 times in a minute. Therefore, the power of the heart is equal to $Power=\dfrac{75PV}{t}$ …. (iv)
According to the given data, $P=10cmHg=13.3\times {{10}^{3}}Pa$.
 $V=75c{{m}^{3}}\\
\Rightarrow V =75\times {{\left( {{10}^{-2}}m \right)}^{3}}\\
\Rightarrow V =75\times {{10}^{-6}}{{m}^{3}}$
And $t=1\min =60s$
Substitute the values of P, V and t in equation (iv).
$\therefore Power=\dfrac{13.3\times {{10}^{3}}\times 75\times {{10}^{-6}}}{60}\approx 1.25W$
Therefore, the power of the heart is equal to 1.25 W

Hence, the correct option is A.

Note: If you do not know that a pressure of 10 cm of Hg is equal to $13.3\times {{10}^{3}}Pa$, then you can calculate the value by using the formula $P=\rho gh$, where $\rho $ is density of mercury, g is acceleration due to gravity and h is the height of mercury column.In this case, substitute the values of $g=9.8m{{s}^{-2}},h=10cm=0.1m$