Which of the following dimension represents the pressure?
Try it first: pick an option, then read the explanation.
Asked in SSC JE 24-01-2018 (Evening Shift)
Detailed Explanation
Dimensional analysis in fluid mechanics expresses physical quantities in terms of fundamental dimensions: Mass [M], Length [L], and Time [T].
1. Definition of pressure:
\[ \text{Pressure } (p) = \frac{\text{Force } (F)}{\text{Area } (A)} \]
2. Dimensions of Force from Newton's second law (\(F = m \cdot a\)):
\[ [F] = [\text{Mass}] \times [\text{Acceleration}] = [M] \times [L T^{-2}] = [M L T^{-2}] \]
3. Dimensions of Area:
\[ [A] = [L^2] \]
4. Divide Force dimensions by Area dimensions:
\[ [p] = \frac{[M L T^{-2}]}{[L^2]} = [M L^{-1} T^{-2}] \]
Why “[M L⁻¹ T⁻²]” Is Correct
Because pressure is defined as normal force per unit area, dividing the dimensions of force \([M L T^{-2}]\) by area \([L^2]\) directly yields \([M L^{-1} T^{-2}]\).
Why the Other Options Are Wrong
- A. [M L T⁻²]: [M L T⁻²] is the dimensional formula of Force or Weight.
- C. [L T⁻²]: [L T⁻²] is the dimensional formula of Acceleration (such as acceleration due to gravity g).
- D. [M L⁻³]: [M L⁻³] is the dimensional formula of Mass Density (mass per unit volume, ρ).
Key Point
Dimensions of Pressure and Stress: [M L⁻¹ T⁻²]; SI unit is Pascal (N/m²).
Key Concepts for Students
- Pressure, stress (tensile, compressive, shear), and elastic moduli (E, G, K) all share dimensions [M L⁻¹ T⁻²].
- Dynamic viscosity has dimensions [M L⁻¹ T⁻¹] (Pa·s).
- Kinematic viscosity has dimensions [L² T⁻¹] (m²/s).
Practice more: this is question 19 in Measurement of Pressure MCQs – Part 2. Try the whole set and check every answer instantly.