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To solve this problem, we can use the Hagen-Poiseuille equation for fluid flow in a cylindrical pipe, which describes the pressure drop for a viscous fluid passing through a tube. The formula for pressure drop \(\Delta P\) is given by:

\[
\Delta P = \frac{8 \cdot \mu \cdot L \cdot v}{r^2}
\]

Where:
- \(\Delta P\) is the pressure drop.
- \(\mu\) is the dynamic viscosity of the fluid.
- \(L\) is the length of the tube.
- \(v\) is the average velocity of the fluid.
- \(r\) is the radius of the tube.

Given the values in the problem:
- \(\mu = 3 \, \text{cP} = 3 \times 10^{-3} \, \text{Pa} \cdot \text{s}\)
- \(L = 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m}\)
- \(v = 0.66 \, \text{mm/s} = 0.66 \times 10^{-3} \, \text{m/s}\)
- \(r = 2 \times 10^{-4} \, \text{cm} = 2 \times 10^{-6} \, \text{m}\)

Let's plug in these values and calculate the pressure drop \(\Delta P\):

\[
\Delta P = \frac{8 \cdot (3 \times 10^{-3}) \cdot (1 \times 10^{-3}) \cdot (0.66 \times 10^{-3})}{(2 \times 10^{-6})^2}
\]

Computing this will give us the pressure drop. I'll perform the calculations now.The pressure drop \(\Delta P\) in the capillary is approximately \(3960 \, \text{Pa}\) (Pascals).

Would you like more details on the calculations or have any questions?

Here are some related questions to expand your understanding:
1. What is the physical meaning of each term in the Hagen-Poiseuille equation?
2. How does the pressure drop change if the radius of the capillary is halved?
3. What role does viscosity play in determining the pressure drop in a fluid?
4. How would the pressure drop be affected if the length of the capillary were doubled?
5. What are the assumptions behind the Hagen-Poiseuille equation and when do they apply?

**Tip:** Always ensure units are consistent when applying formulas in physics to avoid calculation errors.

¿Cuál es la caída de presión de la sangre cuando pasa por un capilar de 1 mm de longitud y de 2.10^-4 cm de radio si la velocidad de la sangre en el centro del capilar es de 0,66 mm/seg y el coeficiente de viscosidad es de 3 cp?

Learn how to calculate the pressure drop of blood in a capillary using the Hagen-Poiseuille equation. This problem involves fluid dynamics, with key data such as viscosity (3 cP), capillary radius (2 × 10^-4 cm), and blood velocity (0.66 mm/s). Suitable for university-level students studying fluid mechanics or physics.

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