Question

Which of the following expression regarding the unit of coefficicnt of viscosity is not truc? (1) dyne \(\mathrm{cm}^{-2} \mathrm{~s}\) (2) dyne \(\mathrm{cm}^{-2} \mathrm{~s}^{-1}\) (3) \(\mathrm{Nm}^{-2} \mathrm{~s}\) (4) 1 poise \(=10^{-1} \mathrm{Nm}^{-2} \mathrm{~s}\)

Short answer

Option 2: dyne \( \text{cm}^{-2} \text{s}^{-1} \) is not a true unit of the coefficient of viscosity.

Step-by-step solution

- Understand the unit of coefficient of viscosity

The coefficient of viscosity (often represented by \( \eta \)) is a measure of a fluid's resistance to gradual deformation by shear stress or tensile stress. Its units in the SI system are \( \text{Nm}^{-2} \text{s} \) (Pascal-seconds or Pa·s) and in the CGS system are poise.

- Analyze each option

Evaluate each option against the correct units to determine which one is not true:1. dyne \( \text{cm}^{-2} \text{s} \) 2. dyne \( \text{cm}^{-2} \text{s}^{-1} \) 3. \( \text{Nm}^{-2} \text{s} \) 4. 1 poise = \( 10^{-1} \text{Nm}^{-2} \text{s} \)

- Verify each unit

Check each option:1. dyne \( \text{cm}^{-2} \text{s} \) is correct because dyne/cm\textsuperscript{2} is a unit of pressure in CGS and s (seconds) is time.2. dyne \( \text{cm}^{-2} \text{s}^{-1} \) is incorrect because the time should be in the numerator, not the denominator.3. \( \text{Nm}^{-2} \text{s} \) is correct as explained earlier.4. 1 poise = \( 10^{-1} \text{Nm}^{-2} \text{s} \) is correct as 1 poise is equivalent to 0.1 Pa·s.

- Identify the incorrect unit

From the verification in Step 3, the incorrect unit expression is Option 2: dyne \( \text{cm}^{-2} \text{s}^{-1} \).

Key concepts

Coefficient of Viscosity

The coefficient of viscosity, often denoted by the Greek letter \( \eta \), is a fundamental property of fluids. It quantifies the fluid's resistance to gradual deformation due to shear stress or tensile stress. In simpler terms, it measures how 'thick' or 'sticky' a fluid is. Fluids with high viscosity resist motion because their molecular composition gives them a lot of internal friction. For instance, honey has a much higher viscosity than water. This means honey flows more slowly compared to water when poured from a container.

SI and CGS Units

Units are vital in the study of viscosity because they allow us to measure and compare different substances. The International System of Units (SI) and the Centimetre-Gram-Second (CGS) system are two universal systems used. The SI unit for viscosity is the Pascal-second (Pa·s), which is derived from the base units of measurement: Newton (N), meter (m), and second (s).
In the CGS system, viscosity is measured in poise (P), named after the French scientist Jean Léonard Marie Poiseuille. Although SI units are more commonly used in modern science, understanding CGS units is sometimes necessary, especially in older scientific literature.
One Pascal-second is equivalent to 10 poise, making conversions between these units straightforward once you have a grasp of their relationship.

Pascal-second

The Pascal-second (Pa·s) is the SI unit for measuring viscosity. It combines the unit of pressure (Pascal) and time (second). One Pascal is defined as one Newton per square meter (N/m^2). Thus, a Pascal-second essentially means how fluid resists flowing when subjected to a Shear force of one Newton per square meter for one second. This unit is very practical for everyday scientific and engineering calculations.
The formula for dynamic viscosity in SI units is represented as:
\( \frac{\mathrm{N}\cdot \mathrm s}{{\mathrm{m}}^2} \/m^2 \)
This formula helps standardize measurements and ensures scientists and engineers around the world speak the same 'language' when discussing the properties of fluids.

Poise

Poise is the CGS unit for viscosity. Named after Poiseuille, this unit is less commonly used today but is still important for historical and certain specialized scientific contexts. One poise (P) can be defined as the viscosity of a fluid in which a force of one dyne per square centimeter maintains a velocity of one centimeter per second. In equation form, it is written as:
\[ \mathrm{P} = \frac{\mathrm{dyne} \cdot \mathrm s}{{\mathrm{cm}}^2} \/cm^2 \]
Although the CGS system is considered somewhat outdated, understanding it can still be crucial when dealing with academic and historical scientific texts. Remember that 1 poise is equivalent to 0.1 Pa·s, which makes it easy to convert to the SI unit system if needed.