Contents
Physics Topics can be challenging to grasp, but the rewards for understanding them are immense.
What is Meant by the Term Dimensionless Quantity?
The dimension of a physical quantity is its relationship with the seven quantities, each of which has been assigned, by convention, a base unit.
These seven quantities, shown earlier in TABLE-3, are
- length (l),
- mass (m),
- time (t),
- electric current (I),
- thermodynamic temperature (T or θ),
- amount of substance (n), and
- luminous intensity (Iv). The dimension of each of them is expressed as a single symbolic factor, as shown in TABLE-6.
Table-6
Dimensions of the base quantities
A short and compact notation for expressing the dimension of a quantity is as follows:
[length] = L or [l] = L,
which is read as “the dimension of length is L”. Here L is a factor that may have multiplication or division with other similar factors, as and when demanded by the relationships among different physical quantities.
Example:
i) Volume (V): V = length × breadth × height. Breadth and height are quantities equivalent to length; each has a dimension L.
So, the dimension of volume =L × L × L, i.e., [V] = L3.
ii) Density (ρ) : ρ \(=\frac{\text { mass }}{\text { volume }}\). Then, [ρ] \(=\frac{[\text { mass }]}{[\text { volume }]}\)
Now, [mass] = M and [volume] = L3.
So, the dimension of density, i.e., [ρ] = \(\frac{M}{L^3}\) = ML-3.
The dimensions of a few important physical quantities are explained in TABLE-7.
Table-7
Relevant relation-ships, dimensions and SI units of some physical quantities
Dimensions from units:
i) The SI unit of density is kg/m3. This unit itself shows that density is actually \(\frac{\text { mass }}{\text { (length) }^3}\), i.e., its dimension is \(\frac{M}{L^3}\) or ML-3. There are many quantities, like density, for which the dimensions may directly be obtained from the units.
ii) The SI unit of force is newton (N). This derived unit, however, does not show its direct relationship with the base units. To get that, some convenient physical relation is to be used. Here, a useful relation is
force = mass × acceleration
So, the unit of force = unit of mass × unit of acceleration
= kg × m/s2 (in SI)
Then, we know that force actually is \(\frac{\text { mass } \times \text { length }}{(\text { time })^2}\)
Hence its dimension is \(\frac{\mathrm{ML}}{\mathrm{T}^2}\) or MLT-2.
The last example shows that the dimension always clearly relates a derived quantity with the base quantities, whereas its unit may or may not. A useful physical formula is often necessary to get this dimensional relationship.
In essence, the connection of all derived physical quantities with the base ones is explicitly displayed by the dimensions, but not always by the conventional units.
Dimensionless quantities: Some physical quantities are actually ratios of other quantities that have the same dimensions. As a result, the ratio becomes dimensionless.
Example:
i) Angle (θ): The angle subtended by a circular arc at its centre is defined as,
θ \(=\frac{\text { length of the circular arc }}{\text { radius of the circle }}\)
Here, both the length and the radius have the dimension of length, i.e., L. So, the dimension of angle is,
[θ] = \(\frac{\mathrm{L}}{\mathrm{L}}\) = L0 = 1
The dimension 1 actually indicates that angle θ is a dimensionless quantity.
ii) Specific gravity (s): By definition, the specific gravity of the material of a body is,
s \(=\frac{\text { mass of the body }}{\text { mass of an equal volume of water }}\), i.e.,
s \(=\frac{\text { mass of the body }}{\text { mass of an equal volume of water }}\)
So, the dimension of specific gravity, [s] = \(\frac{M}{M}\) = M0 = 1. This means that specific gravity is a dimensionless quantity.
If a quantity is dimensionless, its dimension is written as 1. However, expressions like L0, M0, L0M0T0 etc. are equally valid.
Even a dimensionless quantity may have units. Such units are to be assigned to denote different methods of scaling of the quantity For example, an angle θ is dimensionless; but radian and degree are two popular units for the measurement of θ (there are also other, mostly obsolete, units of angle). They actually correspond to the following scalings:
Angle in radian (θc)
Angle in degree (θ0)
It is to be noted that,
- all real numbers are dimensionless;
- a few physical constants (like π, the ratio between circumference and diameter of any circle) are dimensionless, whereas some others (like velocity of light or gravitational constant)
Physical qunantities of the same dimension: Every physical quantity has a definite dimension. But the converse is not true-a dimensional expression alone cannot identify the corresponding physical quantity There are many examples where different quantities have the same dimension. A few of them are given in the following table.
Table-8
| Dimension | Physical Quantities |
| L | Length, breadth, height, depth, thickness, diameter, radius, perimeter, circumference, distance, displacement |
| T-1 | Frequency, circular frequency, angular velocity, velocity gradient, decay constant |
| ML2T-2 | Work, energy torque |
| ML-1T-2 | Pressure, stress, Young’s modulus, bulk modulus, rigidity modulus, energy density |
| MLT-1 | Momentum, impulse of force |
| LT-2 | Acceleration, gravitational intensity |
| MLT-2 | Force, weight |
| ML2T-1 | Angular momentum, Planck’s constant |
| MT-2 | Surface tension, surface energy |
| 1 (or M0L0T0) | Angle, solid angle, specific gravity, strain refractive index, dielectric constant, poison’s ratio. |