Blood flows through a section of a horizontal artery that is partially blocked by a deposit along the artery wall. As a hemoglobin molecule moves from the narrow region into the wider region, its speed changes from v2 = 0.800 m/s to v1 = 0.422 m/s. What is the change in pressure, P1 - P2, that it experiences? The density of blood is 1060 kg/m3.
Would I start by dividing the two values for v?
In a flow without friction the total pressure as sum of static and dynamic pressure is constant.
p1+(ρ•v1²/2) = p2+(ρ•v2²/2),
p1-p2 = (ρ/2)•( v1² -v2²)=
=(1060/2) •(0.422² - 0.8²) = ...
what units do you use for the answer?
To determine the change in pressure, you need to use Bernoulli's equation, which relates changes in fluid speed to changes in pressure. Bernoulli's equation is expressed as:
P + 1/2 * ρ * v^2 = constant
Where:
P is the pressure
ρ is the density of the fluid (in this case, blood)
v is the velocity of the fluid
To calculate the change in pressure (P1 - P2), we can subtract the Bernoulli's equation for the wider region (P1) from the equation for the narrow region (P2).
P2 - P1 = 1/2 * ρ * v1^2 - 1/2 * ρ * v2^2
Substituting the given values:
v1 = 0.422 m/s
v2 = 0.800 m/s
ρ = 1060 kg/m^3
P2 - P1 = 1/2 * 1060 kg/m^3 * (0.422 m/s)^2 - 1/2 * 1060 kg/m^3 * (0.800 m/s)^2
Now, solve for P2 - P1.
To find the change in pressure, we can use Bernoulli's equation, which relates the pressure, velocity, and height of a fluid in a flowing pipe. The equation is as follows:
P1 + 1/2 ρv1^2 + ρgy1 = P2 + 1/2 ρv2^2 + ρgy2
Where:
P1 and P2 are the pressures at two different points in the pipe,
ρ is the density of the fluid (blood in this case),
v1 and v2 are the velocities at two different points in the pipe,
g is the acceleration due to gravity, and
y1 and y2 are the heights of the two points.
Since we are only interested in the change in pressure (P1 - P2), we can simplify the equation by canceling out the terms involving height (since the artery is horizontal) and rewrite it as:
P1 - P2 = 1/2 ρ(v2^2 - v1^2)
Now let's calculate the change in pressure using the given values:
Density of blood (ρ) = 1060 kg/m^3
Initial velocity (v1) = 0.422 m/s
Final velocity (v2) = 0.800 m/s
P1 - P2 = 1/2 × 1060 kg/m^3 × ((0.800 m/s)^2 - (0.422 m/s)^2)
First, square the velocities:
P1 - P2 = 1/2 × 1060 kg/m^3 × (0.64 m^2/s^2 - 0.178 m^2/s^2)
Then subtract the two values:
P1 - P2 = 1/2 × 1060 kg/m^3 × 0.462 m^2/s^2
Finally, multiply the density and the squared velocity difference:
P1 - P2 = 1/2 × 1060 kg/m^3 × 0.462 m^2/s^2 = 237.396 Pa
Therefore, the change in pressure (P1 - P2) that the hemoglobin molecule experiences is approximately 237.396 Pa (Pascals).