How to Calculate the Dimensions of Viscosity with Examples
The dimensions of viscosity are fundamental in understanding the internal resistance to flow present in fluids. Viscosity quantifies the frictional force that acts between adjacent layers of a fluid moving at different velocities. This property is essential in several areas of physics, especially when analyzing fluid dynamics and related phenomena.
Definition and Physical Meaning of Viscosity
Viscosity is the measure of a fluid's resistance to deformation under shear or tensile stress. It arises due to intermolecular forces and represents how much a fluid opposes flow. The coefficient of viscosity, often denoted by $\eta$, characterizes this resistance quantitatively through specific relationships involving force, area, and velocity gradient.
Mathematical Expression for Viscosity
According to Newton's law of viscous flow, the tangential force $F$ required to maintain a velocity gradient $\dfrac{dv}{dx}$ between two layers of liquid with surface area $A$ is given by:
$F = \eta A \dfrac{dv}{dx}$
In this formula, $F$ is the force applied parallel to the layer, $A$ is the area, $\dfrac{dv}{dx}$ is the velocity gradient perpendicular to the direction of flow, and $\eta$ is the coefficient of viscosity.
Derivation of Dimensional Formula for Viscosity
To find the dimensions of viscosity, the formula $\eta = \dfrac{F}{A \dfrac{dv}{dx}}$ is used. The dimensions of each term are considered as follows:
- Force $(F)$: $[M L T^{-2}]$
- Area $(A)$: $[L^2]$
- Velocity $(v)$: $[L T^{-1}]$
- Distance $(x)$: $[L]$
The velocity gradient $\dfrac{dv}{dx}$ thus has dimensions $[L T^{-1}][L^{-1}] = [T^{-1}]$. Therefore, the denominator $A \dfrac{dv}{dx}$ has dimensions $[L^2][T^{-1}] = [L^2 T^{-1}]$.
Substituting the values, the dimensional formula of viscosity is:
$\eta = \dfrac{[M L T^{-2}]}{[L^2 T^{-1}]} = [M L^{-1} T^{-1}]$
This dimensional formula indicates that viscosity depends linearly on mass, is inversely proportional to length, and inversely proportional to time.
Units of Viscosity in Different Systems
In the SI system, viscosity is measured in pascal-seconds (Pa·s), where $1\, \text{Pa·s} = 1\, \text{N m}^{-2} \text{s}$. In the CGS system, the unit is the poise, where $1\, \text{Poise} = 1\, \text{dyn·s·cm}^{-2}$.
| System | Unit of Viscosity |
|---|---|
| SI | Pascal-second (Pa·s) |
| CGS | Poise |
The conversion between these units is $1\, \text{Pa·s} = 10\, \text{Poise}$.
Discussion of Related Physical Quantities
Viscosity is closely connected to other quantities such as force and velocity. It also plays a key role in analyzing stress in fluids. For more details on related dimensional analysis topics, refer to Dimensions Of Force.
Dimensional Analysis and Application in Physics
The dimensional formula $[M L^{-1} T^{-1}]$ helps in checking the correctness of physical equations involving viscosity and performing unit conversions across systems. For related conversion techniques, review Dimensions Of Density.
Summary Table: Dimensional Formulas
| Quantity | Dimensional Formula |
|---|---|
| Viscosity ($\eta$) | $[M L^{-1} T^{-1}]$ |
| Force ($F$) | $[M L T^{-2}]$ |
| Stress | $[M L^{-1} T^{-2}]$ |
Typical Applications of Viscosity
Viscosity is important in lubrication, hydraulic systems, and biological flows. The dimensional understanding is necessary in solving practical problems, especially for questions in competitive exams.
Significance in Dimensional Analysis
Dimensional analysis involving viscosity assists in verifying equations and formulating new relations in fluid mechanics. For further concepts in this domain, examine Dimensions Of Stress.
Comparison with Other Physical Quantities
The dimensions of viscosity are distinct from those of other fluid properties such as density and speed. Details on these differences can befound in Dimensions Of Speed and Dimensions Of Electric Flux.
FAQs on What Are the Dimensions of Viscosity in Physics?
1. What are the dimensions of viscosity?
Viscosity has the dimensional formula [M L-1 T-1].
Key points:
- M: mass
- L: length
- T: time
This formula relates to the measure of a fluid's resistance to flow, which is crucial in physics and fluid mechanics studies.
2. Define viscosity and its SI unit.
Viscosity is the property of a fluid that causes it to resist the relative motion between its layers.
Key points:
- SI unit: Pascal second (Pa·s) or kg m-1 s-1
- It indicates how 'thick' or 'sticky' a fluid is.
- Important for understanding fluid flow in various scientific and engineering contexts.
3. What is the dimensional formula of coefficient of viscosity?
The dimensional formula of coefficient of viscosity is [M L-1 T-1].
Key Relations:
- Derived from force per unit area and velocity gradient.
- Represents the internal friction in fluid flow.
4. Which unit is commonly used to measure viscosity?
The SI unit of viscosity is the Pascal second (Pa·s), but other units like poise (cgs system) are also used.
Key Point:
- 1 poise = 0.1 Pa·s
5. Explain the physical meaning of viscosity with an everyday example.
Viscosity measures how much a fluid resists flowing.
Everyday Example:
- Honey has higher viscosity than water, so it flows more slowly.
- Oils, syrups, and tar are also examples of viscous fluids.
6. How is the dimensional formula for viscosity derived?
The dimensional formula for viscosity is derived using the formula F = η A (dv/dx).
Steps:
- Force (F): [M L T-2]
- Area (A): [L2]
- Velocity gradient (dv/dx): [T-1]
Combining these:
[η] = [F][A-1][(dv/dx)-1] = [M L-1 T-1]
7. What factors affect the viscosity of a fluid?
The viscosity of a fluid depends on several factors.
- Temperature: Increases in temperature usually decrease viscosity of liquids but increase viscosity of gases.
- Nature of fluid: Molecular structure and intermolecular forces affect viscosity.
- Impurities: Presence of dissolved substances can change viscosity.
8. Is viscosity a scalar or vector quantity?
Viscosity is a scalar quantity.
Key points:
- It has magnitude but no direction.
- Represents resistance to flow in a fluid.
9. What is the difference between dynamic and kinematic viscosity?
Dynamic viscosity measures a fluid’s resistance to flow, while kinematic viscosity is the ratio of dynamic viscosity to density.
Key Differences:
- Dynamic viscosity (η): SI unit is Pa·s; dimensional formula is [M L-1 T-1].
- Kinematic viscosity (ν): SI unit is m2/s; dimensional formula is [L2 T-1].
10. State any two applications of viscosity in everyday life.
Viscosity has many practical applications in daily life.
Examples:
- Lubricating oils in machinery reduce wear by managing viscosity.
- Blood viscosity is important for proper circulation in the human body.
