Understanding Dimensionless Quantities

In physics and engineering, a dimensionless quantity is a quantity without any physical unit associated with it. It is a pure number. Such quantities are often defined as ratios of two quantities with the same units, causing the units to cancel out. Let's examine the given options to determine which one is dimensionless.

Specific Gravity

Specific gravity is a dimensionless quantity. It is defined as the ratio of the density of a substance to the density of a reference substance. For liquids and solids, the reference substance is typically water at a specified temperature (often 4 °C), while for gases, it is usually air at a specified temperature and pressure.

The formula for specific gravity (\(SG\)) is: \[ SG = \frac{\text{Density of substance}}{\text{Density of reference substance}} \] Since density has the unit of mass per unit volume (e.g., \( \text{kg/m}^3 \) or \( \text{g/cm}^3 \)), both the numerator and the denominator have the same units. When these units are divided, they cancel out, leaving a pure number with no dimensions. For example: Dimension of density \( = \text{M}\text{L}^{-3} \) Dimension of specific gravity \( = \frac{\text{M}\text{L}^{-3}}{\text{M}\text{L}^{-3}} = \text{M}^0\text{L}^0\text{T}^0 = 1 \) (Dimensionless)

Specific Speed

Specific speed (\( N_s \)) is a dimensionless parameter used to characterize turbomachinery (like pumps, turbines, and fans). It relates the speed, flow rate, head, or power of a machine. While sometimes defined in a way that appears dimensionless for certain units, its fundamental dimensions are not unity. It is typically expressed in specific units depending on the industry and application (e.g., RPM for speed, GPM for flow, feet for head).

For example, for a pump, specific speed can be given by: \[ N_s = \frac{N \sqrt{Q}}{H^{3/4}} \] Where \( N \) is rotational speed, \( Q \) is flow rate, and \( H \) is head. Let's consider the dimensions: Dimension of speed \( N = \text{T}^{-1} \) Dimension of flow rate \( Q = \text{L}^3\text{T}^{-1} \) Dimension of head \( H = \text{L} \) Dimension of specific speed \( N_s = \frac{\text{T}^{-1} \sqrt{\text{L}^3\text{T}^{-1}}}{\text{L}^{3/4}} = \frac{\text{T}^{-1} (\text{L}^3\text{T}^{-1})^{1/2}}{\text{L}^{3/4}} = \frac{\text{T}^{-1} \text{L}^{3/2}\text{T}^{-1/2}}{\text{L}^{3/4}} \] \[ = \text{L}^{(3/2 - 3/4)} \text{T}^{(-1 - 1/2)} = \text{L}^{3/4} \text{T}^{-3/2} \] As seen from the dimensional analysis, specific speed is not dimensionless.

Specific Volume

Specific volume is defined as the volume per unit mass of a substance. It is the reciprocal of density.

The formula for specific volume (\( v \)) is: \[ v = \frac{\text{Volume}}{\text{Mass}} = \frac{1}{\text{Density}} \] The unit of specific volume is typically \( \text{m}^3/\text{kg} \) or \( \text{ft}^3/\text{lb} \). Dimension of specific volume \( = \frac{\text{L}^3}{\text{M}} = \text{L}^3\text{M}^{-1} \) Thus, specific volume is not dimensionless.

Specific Weight

Specific weight (\( \gamma \)) is defined as the weight per unit volume of a substance. It is also known as the unit weight.

The formula for specific weight is: \[ \gamma = \frac{\text{Weight}}{\text{Volume}} = \frac{\text{Mass} \times \text{Acceleration due to gravity}}{\text{Volume}} \] The unit of specific weight is typically \( \text{N/m}^3 \) or \( \text{lb}_f/\text{ft}^3 \). Dimension of weight \( = \text{M}\text{L}\text{T}^{-2} \) (Force) Dimension of volume \( = \text{L}^3 \) Dimension of specific weight \( = \frac{\text{M}\text{L}\text{T}^{-2}}{\text{L}^3} = \text{M}\text{L}^{-2}\text{T}^{-2} \) Therefore, specific weight is not dimensionless.

Conclusion on Dimensionless Nature

Based on the definitions and dimensional analysis, only specific gravity is a dimensionless quantity because it is a ratio of two quantities with identical dimensions, causing all units to cancel out. The other specific quantities (specific speed, specific volume, and specific weight) all possess distinct physical dimensions.