Question

Match List-I with List-II. 

List-I  (A) Coefficient of viscosity  (B) Intensity of wave  (C) Pressure gradient  (D) Compressibility

List-II  (I) [ML-1T-1]  (II) [MT-3]  (III) [ML-2T-2]  (IV) [M-1LT2]

• A. (A)–(I), (B)–(IV), (C)–(III), (D)–(I)
• B. (A)–(I), (B)–(II), (C)–(III), (D)–(IV)
• C. (A)–(IV), (B)–(II), (C)–(III), (D)–(I)
• D. (A)–(IV), (B)–(I), (C)–(II), (D)–(III)

Hint:

In dimensional analysis, the dimensions of physical quantities are crucial for understanding their relationships. Pay attention to how exponents are used to represent various physical properties.

Answer 1

The Correct Option is B

Determine the dimensional formulas for the physical quantities in List-I and match them with the corresponding dimensions in List-II.

Concept Used:

Dimensional analysis expresses physical quantities using fundamental dimensions: Mass (M), Length (L), and Time (T). Dimensions are derived from the definition or formula of a quantity.

Key dimensional formulas:

• Force \( [F] = [MLT^{-2}] \)
• Area \( [A] = [L^2] \)
• Pressure \( [P] = [ML^{-1}T^{-2}] \)
• Energy/Work \( [E] = [ML^2T^{-2}] \)
• Power \( [\text{Power}] = [ML^2T^{-3}] \)

Step-by-Step Solution:

Step 1: (A) Coefficient of viscosity (\( \eta \))

From Newton's law of viscosity, \( F = \eta A \frac{dv}{dx} \), the dimensions of \( \eta \) are:

\[ [\eta] = \frac{[F]}{[A] \cdot [\frac{dv}{dx}]} = \frac{[MLT^{-2}]}{[L^2] \cdot [T^{-1}]} = [ML^{-1}T^{-1}] \]

Matches with (I) in List-II.

Step 2: (B) Intensity of wave (I)

Wave intensity is power per unit area:

\[ [I] = \frac{[\text{Power}]}{[\text{Area}]} = \frac{[ML^2T^{-3}]}{[L^2]} = [MT^{-3}] \]

Matches with (II) in List-II.

Step 3: (C) Pressure gradient

Pressure gradient is the rate of change of pressure with distance:

\[ [\text{Pressure Gradient}] = \frac{[\text{Pressure}]}{[\text{Distance}]} = \frac{[ML^{-1}T^{-2}]}{[L]} = [ML^{-2}T^{-2}] \]

Matches with (III) in List-II.

Step 4: (D) Compressibility (K)

Compressibility is the reciprocal of Bulk Modulus (B), which has the same dimensions as pressure:

\[ [K] = \frac{1}{[\text{Bulk Modulus}]} = \frac{1}{[\text{Pressure}]} = \frac{1}{[ML^{-1}T^{-2}]} = [M^{-1}LT^{2}] \]

Matches with (IV) in List-II.

Final Result:

The correct matching is:

• (A) Coefficient of viscosity → (I) \( [ML^{-1}T^{-1}] \)
• (B) Intensity of wave → (II) \( [MT^{-3}] \)
• (C) Pressure gradient → (III) \( [ML^{-2}T^{-2}] \)
• (D) Compressibility → (IV) \( [M^{-1}LT^{2}] \)

The final matching is (A)-(I), (B)-(II), (C)-(III), (D)-(IV).