In the post dated 30^{th} April 2007, a Matrix Match Type question based on topics in Modern Physics, which appeared in the IIT-JEE 2007 question paper, was discussed. Let us discuss now the question (which appeared in the same examination) based on units and dimensions. Here is the question:

Some physical quantities are given in** column I **and some possible SI units in which these quantities may be expressed are given in **Column II**. Match the physicsl quantities in **column I** with the units in **Column II **and indicate your answer by darkening appropriate bubbles in the 4×4 matrix given in the ORS.

**column I column II**

(A) *GM _{e}M_{s}*

_{ }(p) (volt) (coulomb) (metre)

* G* – universal gravitational constant,

*M _{e}* – mass of the earth,

*M _{s}*– mass of the sun

(B) *3RT / M* (q) (kilogram)(metre)^{3}(second)^{–2}^{}

*R *– universal gas constant,

*T *– absolute temperature,

* M *– molar mass

(C) *F ^{2}/q^{2}B^{2}* (r) (metre)

^{2}(second)

^{–2}

*F *– force,

*q* – charge,

*B* – magnetic field

(D) *GM _{e}/R_{e}* (s) (farad) (volt)

^{2}(kg)

^{–1}

*G *– universal gravitational constant,

*M _{e}* – mass of the earth,

*R _{e}* – radius of the earth

(A) has to be matched with (p) and (q). This can be done in various ways. But it will be convenient to do it by noting that the quantity GM_{e}M_{s }is force×distance^{2} since the gravitational force between the earth and the sun is GM_{e}M_{s}/r^{2}, with usual notations.

But, force×distance^{2} = energy×distance. The quantity (volt) (coulomb) (metre) also is energy×distance (since ‘Vq’ is energy). The quantity (kilogram)(metre)^{3}(second)^{–2} also is energy×distance (remembering F = ma and energy = Fs).

(B) has to be matched with (r) and (s). This can be proved by noting that 3RT / M is the square of the r.m.s. velocity of molecules. Its unit is therefore (metre)^{2}(second)^{–2}.

The quantity (farad) (volt)^{2}(kg)^{–1} also has the dimensions of velocity^{2} since farad metre^{2} has the dimensions of energy (remember E = ½ CV^{2}) and energy per kg has dimensions of velocity^{2} (E = p^{2}/2m so that E/m = p^{2}/2m^{2} = m^{2}v^{2}/2m^{2} which has dimensions of velocity^{2}).

(C) is to be matched with (r) and (s) since F^{2}/q^{2}B^{2} has dimensions of velocity^{2}. (Remember the expression F = qvB for magnetic force on a moving charge). ^{}

(D) is to be matched with (r) and (s) since* *GM_{e}/R_{e} is gR_{e} which has dimensions of [(metre/sec^{2}) ×metre] and therefore those of velocity^{2}.

It will be difficult for you to remember the dimensions of all the physical quantities, especially those you come across in branches of physics other than mechanics. But you will remember the important expressions you derive in all the branches. The Matrix Match Type question we discussed is aimed at checking your knowledge of those expressions and your understanding of the significance of dimensions.

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