Question: Which of the following pair does not have similar dimensions? [PMT/NEET-2001]
a. Stress and pressure
b. Angle and strain
c. Tension and surface tension
d. Planck’s constant and angular momentum
The correct answer is (c) Tension and surface tension.
These two quantities have completely different dimensions, making them the pair that does not have similar dimensions.
Dimensional Analysis
Let’s analyze each option systematically:
a. Stress and Pressure
- Stress = Force/Area = $[MLT^{-2}]/[L^2] = [ML^{-1}T^{-2}]$
- Pressure = Force/Area = $[MLT^{-2}]/[L^2] = [ML^{-1}T^{-2}]$
- Result: Same dimensions ✓
b. Angle and Strain
- Angle = Arc length/Radius = $[L]/[L] = [M^0L^0T^0]$ (dimensionless)
- Strain = Change in length/Original length = $[L]/[L] = [M^0L^0T^0]$ (dimensionless)
- Result: Same dimensions ✓
c. Tension and Surface Tension
- Tension = Force = $[MLT^{-2}]$
- Surface tension = Force per unit length = $[MLT^{-2}]/[L] = [MT^{-2}]$
- Result: Different dimensions ✗
d. Planck’s Constant and Angular Momentum
- Planck’s constant (h) = Energy × Time = $[ML^2T^{-2}] × [T] = [ML^2T^{-1}]$
- Angular momentum = $mvr = [M][LT^{-1}][L] = [ML^2T^{-1}]$
- Result: Same dimensions ✓
Key Points to Remember
- Tension is simply a pulling force measured in Newtons (N)
- Surface tension is force per unit length, measured in N/m or dyne/cm
- Remember that surface tension creates a “skin” effect on liquid surfaces due to intermolecular forces
- This dimensional difference explains why surface tension has units like N/m while tension has units of N
Memory Tip
Think of surface tension as the “strength” of the liquid’s surface per unit length, while tension is just the total pulling force – this fundamental difference in their physical meanings leads to different dimensions.
