The drawing shows an intravenous feeding. With the distance shown, nutrient solution (ρ = 1050 kg/m3) can just barely enter the blood in the vein. What is the gauge pressure of the venous blood? Express your answer in millimeters of mercury.
To determine the gauge pressure of the venous blood, we need to apply Bernoulli's equation, which relates the pressure, velocity, and height of a fluid.
First, let's identify the known and unknown quantities in the problem:
Known:
- Density of the nutrient solution (ρ) = 1050 kg/m^3
- Height of the fluid column (h) = distance shown in the drawing
Unknown:
- Gauge pressure of the venous blood
Now, let's write down Bernoulli's equation:
P + (1/2)ρv^2 + ρgh = constant
Where:
- P is the pressure of the fluid
- ρ is the density of the fluid
- v is the velocity of the fluid
- g is the acceleration due to gravity
- h is the height of the fluid column
In this case, we can assume that the velocity of the fluid (v) is negligible since it is not mentioned in the problem.
Therefore, we can simplify Bernoulli's equation to:
P + ρgh = constant
Since the fluid in the vein is in equilibrium, the pressure inside the vein (P) is equal to atmospheric pressure. Hence, we have:
P = Patm
Substituting these values into the equation, we get:
Patm + ρgh = constant
To find the gauge pressure of the venous blood, we need to rearrange the equation:
Patm = constant - ρgh
Now, let's calculate the gauge pressure in terms of millimeters of mercury (mmHg). We know that 1 atm = 760 mmHg, so we have:
Patm (mmHg) = (constant - ρgh) / 760
To compute the value of constant, we need additional information from the problem. If the problem provides more information, please provide it so that we can proceed with the calculation.