Question

The C.G.S. unit of coefficient of viscosity is poise and the SI unit is \(\mathrm{Pa}-\mathrm{S}\). What is the relation between the two ? (A) \(1 \mathrm{~Pa}-\mathrm{s}=10^{3}\) poise (B) \(1 \mathrm{~Pa}-\mathrm{s}=10^{4}\) poise (C) \(1 \mathrm{~Pa}-\mathrm{s}=10\) poise (D) \(1 \mathrm{~Pa}-\mathrm{s}=100\) poise

Short answer

(C) \(1\ \mathrm{Pa}-\mathrm{s} = 10\) poise.

Step-by-step solution

Understand the units and their differences

In the SI (International System of Units) system, the unit of coefficient of viscosity is Pascal-second (Pa-s). In the CGS (Centimeter-Gram-Second) system, the unit is poise. The conversion between these two units can be found by understanding the units of force involved in CGS and SI. In SI, the unit of force is Newton (N), whereas in CGS, the unit of force is dyne.

Determine the conversion factor between force units

We need to find the conversion factor between Newtons (N, the SI unit of force) and dynes (dyn, the CGS unit of force). We know that 1 N is equal to \(10^5\) dynes. Thus, the conversion factor between the two is: \[1~\mathrm{N} = 10^5 ~\mathrm{dyn}.\]

Convert Pa-s to poise through dimensional analysis

Using dimensional analysis, we'll convert Pascal-seconds (Pa-s) to poise. A Pascal is defined as 1 N/m², and since 1 N is equal to \(10^5\) dynes, we have: \[1 ~\mathrm{Pa} = 1 ~\frac{\mathrm{N}}{\mathrm{m^2}} = 1 ~\frac{10^5 ~\mathrm{dyn}}{(\mathrm{m})^2}.\] To convert Pa-s to poise, knowing 1 m = 100 cm, we can write: \[1 ~\mathrm{Pa} = 1 ~\frac{10^5 ~\mathrm{dyn}}{(\mathrm{m})^2} = 1 ~\frac{10^5 ~\mathrm{dyn}}{(100\ \mathrm{cm})^2} = \frac{10 ~\mathrm{dyn}}{\mathrm{cm^2}}.\] Therefore, we have: \[1\ \mathrm{Pa}-\mathrm{s} = 1 ~\frac{\mathrm{Pa}}{\mathrm{s}} = 1 ~\frac{10 ~\mathrm{dyn}}{\mathrm{s}\cdot\mathrm{cm^2}}.\] Recalling that 1 poise is equal to 1 (dyne)/(s·cm²), we can simplify the expression above to find the conversion factor: \[1\ \mathrm{Pa}-\mathrm{s} = 10 ~\frac{\mathrm{dyn}}{\mathrm{s}\cdot\mathrm{cm^2}} = 10 ~\mathrm{poise}.\] Thus, the correct answer is: (C) \(1\ \mathrm{Pa}-\mathrm{s} = 10\) poise.

Key concepts

CGS system

The CGS system, or Centimeter-Gram-Second system, is a metric system of units that dates back to the 19th century. It uses centimeters for length, grams for mass, and seconds for time. This system is often employed in physics and engineering as it simplifies solving problems by relying on smaller-scale units. For example, the unit of force in the CGS system is the dyne, where 1 dyne is the force required to accelerate a one-gram mass by one centimeter per square second. In terms of viscosity, the unit in the CGS system is the poise. One poise is defined as one dyne-second per square centimeter. Despite its simplicity, the CGS system's popularity has waned since the adoption of the SI units, owing to the uniformity and practicality offered by the latter across different scientific and industrial fields.

SI units

The International System of Units (SI units) is the most widely used set of measurement units in the world today, offering a standardized form that facilitates global communication in science and technology. In SI, the unit of force is the Newton, which is defined as the force needed to accelerate a mass of one kilogram by one meter per square second. One of the primary reasons for SI's dominance is its scalability and ease of conversion between different units of the metric system. For measuring viscosity, the SI unit is the Pascal-second (Pa-s). The Pascal, a unit of pressure, is equivalent to one Newton per square meter. Therefore, a Pascal-second corresponds to the viscosity of a fluid exerting one Pascal of shear stress while flowing at a velocity gradient of one meter per second over a distance of one meter. The broad acceptance of SI units makes them indispensable for accurate scientific and industrial applications.

Dimensional analysis

Dimensional analysis is an essential technique in physics and engineering that allows for the converting and checking of consistency across different measurement units, especially between systems like CGS and SI. By using the fundamental dimensions of a quantity—such as length (L), mass (M), and time (T)—we can ensure that equations and calculations remain consistent regardless of the units used. For example, to convert viscosity from SI units (Pascal-seconds) to CGS units (poise), one would analyze the formulation by considering the units of force (Newtons and dynes). Using the conversion factor from 1 Newton to 10^5 dynes, we verify the units' consistency from SI to CGS by expressing them in terms of these fundamental dimensions. This ensures that calculations remove potential errors due to mismatched units. Dimensional analysis is a powerful tool for both verifying equation accuracy and converting complex formulas across unit systems.

Pascal-second

The Pascal-second (Pa-s) is the SI unit used to measure the viscosity of a fluid, which is akin to the thickness or resistance of the fluid as it flows. One Pascal is defined as the pressure applied with one Newton of force over an area of one square meter. When it comes to describing how a fluid resists deformation, the reciprocal relationship between shear stress and strain rate is expressed in Pascal-seconds. A larger Pascal-second value indicates a thicker fluid, one that resists flow more strongly. The introduction of Pa-s enables professionals across various disciplines to maintain uniformity in measuring viscosity, contributing to better communication and comparability of scientific data globally.

Poise

Poise is the unit of dynamic viscosity measurement in the CGS system, honoring the French physicist Jean Léonard Marie Poiseuille. One poise represents a viscosity of one dyne-second per square centimeter. When compared to the SI unit Pascal-second (Pa-s) for viscosity, one Pa-s is equal to 10 poises. This conversion recognizes that 1 Newton equals 10^5 dynes and adjusts for the differences in length dimensions used between meters and centimeters. The use of poise, though more traditional, provides a convenient scale for substances with moderate viscosity and finds occasional use in specific industrial and research contexts due to its practicality in particular fields.