Question:
Which of the following is dimensionless?
Which of the following is dimensionless?
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Any physical property defined as a ratio of identical properties is dimensionless.
Common examples in dairy engineering include specific gravity, refractive index, and fluid flow ratios like the Reynolds and Nusselt numbers.
Common examples in dairy engineering include specific gravity, refractive index, and fluid flow ratios like the Reynolds and Nusselt numbers.
Updated On: Jul 23, 2026
- Volume
- Density
- Specific gravity
- Surface tension
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Concept:
Physical quantities are characterized by their dimensions, which are expressed in terms of fundamental dimensions like Mass (\(\text{M}\)), Length (\(\text{L}\)), and Time (\(\text{T}\)).
A dimensionless quantity is a ratio of two physical quantities with identical dimensions, resulting in a net numerical value without any units.
Step 2: Detailed Explanation:
Let us analyze the dimensions of each option to evaluate which is dimensionless:
- (A) Volume:
Volume is the three-dimensional space occupied by a substance, measured in cubic meters (\(\text{m}^3\)).
Its dimensional formula is:
\[ [\text{Volume}] = [\text{L}^3] \]
- (B) Density:
Density is mass per unit volume, measured in kilograms per cubic meter (\(\text{kg/m}^3\)).
Its dimensional formula is:
\[ [\text{Density}] = [\text{M L}^{-3}] \]
- (C) Specific gravity:
Specific gravity (or relative density) is the ratio of the density of a substance to the density of a reference substance (typically pure water at 4°C):
\[ \text{Specific Gravity} = \frac{\rho_{\text{substance}}}{\rho_{\text{water}}} \]
Because it is the ratio of two identical physical quantities, the units and dimensions cancel out completely:
\[ [\text{Specific Gravity}] = \frac{[\text{M L}^{-3}]}{[\text{M L}^{-3}]} = [\text{M}^0 \text{L}^0 \text{T}^0] \]
Therefore, specific gravity is a dimensionless quantity.
- (D) Surface tension:
Surface tension is force per unit length, measured in Newtons per meter (\(\text{N/m}\)).
Its dimensional formula is:
\[ [\text{Surface Tension}] = \frac{[\text{M L T}^{-2}]}{[\text{L}]} = [\text{M T}^{-2}] \]
Step 3: Final Answer:
Specific gravity is a dimensionless physical quantity.
Physical quantities are characterized by their dimensions, which are expressed in terms of fundamental dimensions like Mass (\(\text{M}\)), Length (\(\text{L}\)), and Time (\(\text{T}\)).
A dimensionless quantity is a ratio of two physical quantities with identical dimensions, resulting in a net numerical value without any units.
Step 2: Detailed Explanation:
Let us analyze the dimensions of each option to evaluate which is dimensionless:
- (A) Volume:
Volume is the three-dimensional space occupied by a substance, measured in cubic meters (\(\text{m}^3\)).
Its dimensional formula is:
\[ [\text{Volume}] = [\text{L}^3] \]
- (B) Density:
Density is mass per unit volume, measured in kilograms per cubic meter (\(\text{kg/m}^3\)).
Its dimensional formula is:
\[ [\text{Density}] = [\text{M L}^{-3}] \]
- (C) Specific gravity:
Specific gravity (or relative density) is the ratio of the density of a substance to the density of a reference substance (typically pure water at 4°C):
\[ \text{Specific Gravity} = \frac{\rho_{\text{substance}}}{\rho_{\text{water}}} \]
Because it is the ratio of two identical physical quantities, the units and dimensions cancel out completely:
\[ [\text{Specific Gravity}] = \frac{[\text{M L}^{-3}]}{[\text{M L}^{-3}]} = [\text{M}^0 \text{L}^0 \text{T}^0] \]
Therefore, specific gravity is a dimensionless quantity.
- (D) Surface tension:
Surface tension is force per unit length, measured in Newtons per meter (\(\text{N/m}\)).
Its dimensional formula is:
\[ [\text{Surface Tension}] = \frac{[\text{M L T}^{-2}]}{[\text{L}]} = [\text{M T}^{-2}] \]
Step 3: Final Answer:
Specific gravity is a dimensionless physical quantity.
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