Which one of the following is a dimensionless quantity? (a) mass (b) weight (c) specific weight (d) Reynolds number
The correct answer and explanation is:
The correct answer is (d) Reynolds number.
Explanation:
A dimensionless quantity is one that does not have any physical dimensions or units. It is a pure number obtained by dividing two quantities with the same units, so the units cancel out. Now, let’s look at each option:
- Mass: Mass is a physical quantity that measures the amount of matter in an object. It has the dimension of mass (M) and is typically measured in kilograms (kg), which makes it a dimensional quantity. Therefore, mass is not dimensionless.
- Weight: Weight is the force exerted by gravity on an object and is calculated as the product of the mass of an object and the acceleration due to gravity (W = m * g). Since weight is a force, it has dimensions of mass times acceleration, which corresponds to the dimension [M L T⁻²] (where M is mass, L is length, and T is time). Hence, weight is not dimensionless.
- Specific weight: Specific weight is the weight per unit volume of a substance. It is calculated by dividing the weight of an object by its volume. The formula is γ=WV\gamma = \frac{W}{V}, where WW is the weight and VV is the volume. Since weight is dimensionful and volume has the dimension of length cubed, specific weight is not dimensionless.
- Reynolds number: Reynolds number is used to predict the flow regime of a fluid (laminar or turbulent) and is defined as the ratio of inertial forces to viscous forces. The formula for Reynolds number is: Re=ρuLμRe = \frac{\rho u L}{\mu} where ρ\rho is the fluid density, uu is the fluid velocity, LL is a characteristic length, and μ\mu is the dynamic viscosity. The units of density, velocity, length, and viscosity cancel out, leaving Reynolds number as a dimensionless quantity. This makes it the correct choice.
Thus, Reynolds number is the only option that does not depend on physical dimensions and is dimensionless.