Question

Match the parameters in Group I with their corresponding dimensions in Group II.

Group I		Group II
P	Shear modulus	1	\(M^\circ L^\circ T^\circ\)
Q	Hydraulic conductivity	2	\(M^1L^{-3}T^\circ\)
R	Volumetric strain	3	\(M^0 L^1 T^{-1}\)
S	Dry density	4	\(M^1 L^{-1} T^{-2}\)

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

Answer 1

The Correct Option is A

To match the parameters in Group I with their corresponding dimensions in Group II, we start by identifying the dimensions of each parameter:

1. Shear Modulus (P): Shear modulus is defined as the ratio of shear stress to shear strain. Shear stress has dimensions of force per unit area, which is \(ML^{-1}T^{-2}\), while strain is dimensionless. Thus, shear modulus has dimensions \(ML^{-1}T^{-2}\).
2. Hydraulic Conductivity (Q): This parameter measures the ease with which a fluid can move through pore spaces or fractures in a permeable medium. The dimension is typically represented as velocity, or \(LT^{-1}\), corresponding to \(M^0L^1T^{-1}\) since it relates to the volume of water that crosses a unit area per unit time.
3. Volumetric Strain (R): Volumetric strain is a measure of deformation representing the relative volume change of a material. Since it is a ratio of change in volume to original volume, it is dimensionless, or \(M^0L^0T^0\).
4. Dry Density (S): Dry density is the mass of solids per unit volume of soil. Hence, the dimensions are \(ML^{-3}\).

Given the dimensions in Group II:

• \(M^\circ L^\circ T^\circ\) corresponds to volumetric strain, which is dimensionless.
• \(M^1L^{-3}T^\circ\) matches dry density, \(ML^{-3}\).
• \(M^0 L^1 T^{-1}\) fits hydraulic conductivity, similar to velocity.
• \(M^1 L^{-1} T^{-2}\) is for shear modulus as calculated for stress.

Therefore, the correct matching will be:

• P-4: Shear Modulus - \(M^1 L^{-1} T^{-2}\)
• Q-3: Hydraulic Conductivity - \(M^0 L^1 T^{-1}\)
• R-1: Volumetric Strain - \(M^\circ L^\circ T^\circ\)
• S-2: Dry Density - \(M^1L^{-3}T^\circ\)

Hence, the correct option is: P-4, Q-3, R-1, S-2.