I believe in standardizing automobiles, not human beings

– Albert Einstein

In most of the entrance examination question papers you will find at least one question on dimensions of physical quantities. Here are a few typical multiple choice questions in this section:

**(1)** Which one among the following quantities is a dimensional constant?

(a) Dielectric constant of water

(b) Speed of light in free space

(c) Viscosity of water

(d) Ratio of specific heats of a diatomic gas

(e) Reynolds number

Options (a) and (c) are not constants since the dielectric constant and viscosity depend on other parameters. Options (d) and (e) are dimensionless numbers.

[Reynolds number (used in assessing the turbulence of fluids) is the ratio of inertial force to force of viscosity]

The correct option is the *speed of light in free space* which is a fundamental constant with dimensions LT^{–1}.

**(2)** If *F* denotes force and *t* time, then in the equation* F* = *at*^{–1} + *bt*^{2}, the dimensions of *a* and *b* respectively are

(a) LT^{–4} and LT^{–1}

(b) LT^{–1} and LT^{–4}

(c) MLT^{–4 }and MLT^{–1}

(d) MLT^{–1} and MLT^{–4}

(e) MLT^{–3} and MLT^{–2}

The above question appeared in Kerala Engineering Entrance ((KEAM) 2011 question paper.

The dimensions of *at*^{–1} and *bt*^{2} have to be that of force which is MLT^{–2}. Therefore the dimensions of *a* must be MLT^{–1} and the dimensions of *b* must be MLT^{–4}.

The correct option is (d).

The following question appeared in Karnataka CET 2004 question paper:

**(3)** The physical quantity having the same dimensions as Planck’s constant *h *is

(1) Boltzmann constant

(2) force

(3) linear momentum

(4) angular momentum

Most of you will remember that the unit of *h* is joule second. Therefore *h *has the dimensions of the product of work and time which is ML^{2}T^{–2}×T = ML^{2}T^{–1}.

Angular momentum is the moment of linear momentum and has dimensions L×MLT^{–1} = ML^{2}T^{–1}.

The correct option is (4).

The following question appeared in EAMCET 2008 Engineering Entrance Exam question paper. You will answer it in no time if you remember that the dimensions of Planck’s constant are those of *angular momentum*.

**(4**) The energy (*E*), angular momentum (*L*) and universal gravitational constant (*G*) are chosen as fundamental quantities. The dimensions of universal gravitational constant in the dimensional formula for Planck’s constant (*h*) is

(a) zero

(b) – 1

(c) 5/3

(d) 1

Since the dimensions of Planck’s constant are those of angular momentum, it follows that the dimensions of universal gravitational constant in the dimensional formula for Planck’s constant (*h*) is zero.

[In terms of *E*, *L* and *G* which are assumed as fundamental quantities in the above question, Planck’s constant (*h*) has zero dimension in *E*, one dimension in *L* and zero dimension in *G*].