Question

Match the following as per dimensional formula.

• A. 1-R, 2-P, 3-Q, 4-S
• B. 1-S, 2-Q, 3-R, 4-S
• C. 1-R, 2-P, 3-S, 4-Q
• D. 1-S, 2-Q, 3-Q, 4-S

Hint:

Remember these shortcuts: Pressure \(\rightarrow\) Force/Area, Surface tension \(\rightarrow\) Force/Length, Surface energy \(\rightarrow\) Energy.

Answer 1

The Correct Option is A

Concept:
Dimensional formulas express a physical quantity in terms of fundamental quantities mass \((M)\), length \((L)\), and time \((T)\).
Step 1: Write Dimensional Formulas of Each Quantity
Pressure:
\[ \text{Pressure} = \frac{\text{Force}}{\text{Area}} = \frac{M L T^{-2}}{L^{2}} = M L^{-1} T^{-2} \] So, Pressure \(\rightarrow\) (R) Coefficient of Viscosity:
\[ \eta = \frac{\text{Shearing stress}}{\text{Velocity gradient}} = \frac{M L^{-1} T^{-2}}{T^{-1}} = M L^{-1} T^{-1} \] So, Coefficient of viscosity \(\rightarrow\) (P) Surface Tension:
\[ \text{Surface tension} = \frac{\text{Force}}{\text{Length}} = \frac{M L T^{-2}}{L} = M T^{-2} = M L^{0} T^{-2} \] So, Surface tension \(\rightarrow\) (Q) Surface Energy:
Surface energy is energy per unit area multiplied by area, hence it has the dimensions of energy: \[ \text{Energy} = M L^{2} T^{-2} \] So, Surface energy \(\rightarrow\) (S)
Step 2: Final Matching
\[ (1)\!-\!R,\quad (2)\!-\!P,\quad (3)\!-\!Q,\quad (4)\!-\!S \]