Examkin

Question

Difficulty: MediumFundamental and Derived Units

The coefficient of dynamic viscosity η\eta of a fluid can be defined using Newton's formula for viscous flow, η=FAΔvΔx\eta = \frac{F}{A \frac{\Delta v}{\Delta x}}, where FF is the viscous drag force, AA is the contact surface area, and ΔvΔx\frac{\Delta v}{\Delta x} is the velocity gradient perpendicular to the flow. Which of the following combinations of fundamental SI base units is equivalent to the coefficient of dynamic viscosity?

  1. kg⋅m−1⋅s−1\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-1}Answer
  2. B
    kg⋅m⋅s−2\text{kg}\cdot\text{m}\cdot\text{s}^{-2}
  3. C
    kg⋅m−2⋅s−1\text{kg}\cdot\text{m}^{-2}\cdot\text{s}^{-1}
  4. D
    kg⋅m−1⋅s−2\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}

Answer

The coefficient of dynamic viscosity expressed in fundamental SI base units is kg⋅m−1⋅s−1\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-1}.
The unit of dynamic viscosity is derived from η=FA(Δv/Δx)\eta = \frac{F}{A (\Delta v / \Delta x)}. Substituting base units yields kg⋅m⋅s−2m2⋅s−1=kg⋅m−1⋅s−1\frac{\text{kg}\cdot\text{m}\cdot\text{s}^{-2}}{\text{m}^2 \cdot \text{s}^{-1}} = \text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-1}, which correctly represents the unit in base SI quantities.

Step-by-Step Solution

1
Identify the fundamental SI base units for each quantity in the formula
Force FF has units kg⋅m⋅s−2\text{kg}\cdot\text{m}\cdot\text{s}^{-2}; Area AA has units m2\text{m}^2; Velocity gradient ΔvΔx\frac{\Delta v}{\Delta x} has units m⋅s−1m=s−1\frac{\text{m}\cdot\text{s}^{-1}}{\text{m}} = \text{s}^{-1}.
Decomposing derived physical quantities into SI base units is essential for dimensional reduction.
2
Substitute these fundamental units into the given formula for η\eta
Unit of η=kg⋅m⋅s−2m2⋅s−1\text{Unit of } \eta = \frac{\text{kg}\cdot\text{m}\cdot\text{s}^{-2}}{\text{m}^2 \cdot \text{s}^{-1}}.
Direct algebraic substitution yields the compound unit expression.
3
Simplify the powers of mass, length, and time
kg1⋅m1−2⋅s−2−(−1)=kg⋅m−1⋅s−1\text{kg}^1 \cdot \text{m}^{1 - 2} \cdot \text{s}^{-2 - (-1)} = \text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-1}.
Applying exponent rules simplifies the expression to base units.

Key Concept

Expressing derived units in terms of fundamental SI base units (kilogram, meter, second)
Rate this question

Topics

Fundamental and Derived Units

More questions from this topic