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Chapter 9: Problem 53

Using Poiseuille's law [Eq. (9-15)], show that viscosity has SI units of pascal-seconds.

Short Answer

Expert verified
Answer: The SI units of viscosity are pascal-seconds (Pa·s).

Step by step solution

01

Write down Poiseuille's law equation

The Poiseuille's law equation can be written as: \(Q = \frac{πr^4 ΔP}{8 η L}\), where: - \(Q\) is the volumetric flow rate (m³/s), - \(r\) is the radius of the pipe (m), - \(ΔP\) is the pressure difference between the two ends of the pipe (Pa or N/m²), - \(η\) is the dynamic viscosity of the fluid (Pa·s), - \(L\) is the length of the pipe (m).
02

Express pressure difference in terms of force and area

Recall that pressure is defined as the force exerted per unit area: \(P = \frac{F}{A}\), where \(F\) is the force (N) and \(A\) is the area (m²). Therefore, the pressure difference, \(ΔP\), can be expressed in the same units, as \(ΔP = \frac{ΔF}{A}\) or N/m².
03

Rearrange the equation for viscosity

Now we will solve the Poiseuille's law equation for viscosity. Rearranging the equation, we get: \(η = \frac{πr^4 ΔP}{8 Q L}\).
04

Substitute the SI units for each quantity

Substitute the SI units for each quantity in the equation for viscosity: \(η(\text{Pa·s}) = \frac{π(\text{m})^4 (\text{N/m}²)}{8 (\text{m³/s}) (\text{m})}\).
05

Simplify the expression

Simplify the expression to determine the SI units of viscosity: \(\text{Pa·s} = \frac{\cancel{\text{m}}^4 (\text{N}/\cancel{\text{m}}²)}{8 (\text{m³}/\cancel{\text{s}}) \cancel{\text{m}}}\). Dividing both the numerator and denominator of the equation by \(m³\), we get: \(\text{Pa·s} = \frac{\text{N}·\cancel{\text{m}³}}{\cancel{\text{m}³}} = \text{N·s}/\text{m}²\) Since \(1 \text{Pa} = 1 \text{N/m}²\), we can replace the \(\text{N}/\text{m}²\) with \(\text{Pa}\): \(\text{Pa·s} = \text{Pa·s}\).
06

Conclusion

Using Poiseuille's law, we have shown that the SI units of viscosity are pascal-seconds (Pa·s).

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Most popular questions from this chapter

The volume flow rate of the water supplied by a well is $2.0 \times 10^{-4} \mathrm{m}^{3} / \mathrm{s} .\( The well is \)40.0 \mathrm{m}$ deep. (a) What is the power output of the pump - in other words, at what rate does the well do work on the water? (b) Find the pressure difference the pump must maintain. (c) Can the pump be at the top of the well or must it be at the bottom? Explain.
A 10 -kg baby sits on a three-legged stool. The diameter of each of the stool's round feet is \(2.0 \mathrm{cm} .\) A \(60-\mathrm{kg}\) adult sits on a four-legged chair that has four circular feet, each with a diameter of $6.0 \mathrm{cm} .$ Who applies the greater pressure to the floor and by how much?
A water tower supplies water through the plumbing in a house. A 2.54 -cm- diameter faucet in the house can fill a cylindrical container with a diameter of \(44 \mathrm{cm}\) and a height of \(52 \mathrm{cm}\) in 12 s. How high above the faucet is the top of the water in the tower? (Assume that the diameter of the tower is so large compared to that of the faucet that the water at the top of the tower does not move. \()\)
A shallow well usually has the pump at the top of the well. (a) What is the deepest possible well for which a surface pump will work? [Hint: A pump maintains a pressure difference, keeping the outflow pressure higher than the intake pressure.] (b) Why is there not the same depth restriction on wells with the pump at the bottom?
You are hiking through a lush forest with some of your friends when you come to a large river that seems impossible to cross. However, one of your friends notices an old metal barrel sitting on the shore. The barrel is shaped like a cylinder and is \(1.20 \mathrm{m}\) high and \(0.76 \mathrm{m}\) in diameter. One of the circular ends of the barrel is open and the barrel is empty. When you put the barrel in the water with the open end facing up, you find that the barrel floats with \(33 \%\) of it under water. You decide that you can use the barrel as a boat to cross the river, as long as you leave about $30 \mathrm{cm}$ sticking above the water. How much extra mass can you put in this barrel to use it as a boat?
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