Blood from an animal is placed in a bottle 2.0 m above a 3.4 cm long needle, of inside diameter 0.35 mm, from which it flows at a rate of 2.9 cm^3/min. What is the viscosity of this blood? Assume Pblood}=1.05 * 10^3 kg/m^3.
if you exert a force of 500 N to walk 4 m up a flight of stairs in 4 s, how much power do you use?
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To find the viscosity of the blood, we can use Poiseuille's law, which relates the flow rate of a liquid through a narrow tube to the pressure difference across the tube, the tube length, and the tube radius.
The formula for Poiseuille's Law is as follows:
Q = (π * ΔP * r^4) / (8 * η * L)
Where:
Q is the flow rate of the blood (2.9 cm^3/min)
ΔP is the pressure difference across the tube (which we will calculate)
r is the radius of the needle (0.35 mm / 2)
η is the viscosity of the blood (which we want to find)
L is the length of the needle (3.4 cm)
First, convert the flow rate to SI units:
Q = (2.9 cm^3/min) * (1 mL / 1 cm^3) * (1 L / 1000 mL) * (1 min / 60 s)
Q ≈ 0.00004833 m^3/s
Next, convert the radius of the needle to meters:
r = (0.35 mm / 2) * (1 m / 1000 mm)
r = 0.000175 m
Now, we can rearrange the equation to solve for η:
η = (π * ΔP * r^4) / (8 * Q * L)
Rearranging further to solve for ΔP:
ΔP = (8 * Q * L * η) / (π * r^4)
Plugging in the known values:
ΔP = (8 * 0.00004833 m^3/s * 0.034 m * η) / (π * (0.000175 m)^4)
Dividing the numerator by the denominator:
ΔP = (0.00001237 m^3/s * η) / 0.000006129 m^4
Simplifying:
ΔP ≈ 2.02 η
Now we need to calculate ΔP to find η.
To determine the pressure difference ΔP across the tube, we can use the hydrostatic pressure formula. The pressure at the bottom of the tube is equal to the pressure at the top of the tube plus the pressure due to the height difference:
P2 = P1 + ρ * g * h
Where:
P2 is the pressure at the bottom of the tube (which we will use as ΔP)
P1 is the pressure at the top of the tube (which we assume is atmospheric pressure, approximately 1 atm)
ρ is the density of the blood (1.05 * 10^3 kg/m^3)
g is the acceleration due to gravity (9.8 m/s^2)
h is the height difference (2 m)
Plugging in the values:
ΔP = (1 atm + (1.05 * 10^3 kg/m^3 * 9.8 m/s^2 * 2 m) * (101325 Pa / 1 atm)
ΔP ≈ 25925 Pa
Now we can substitute this value of ΔP into the viscosity equation:
25925 Pa ≈ 2.02 η
Solving for η:
η ≈ 25925 Pa / 2.02
η ≈ 12819.8 Pa s (or kg/(m·s))
Therefore, the viscosity of the blood is approximately 12820 Pa s (or kg/(m·s)).