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 physical 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.

Answer 1

(A) : From, $F=\frac{G{M}_e{M}_s}{R^2}⟹G{M}_e{M}_s=F{R}^2$

The S.I unit of $F$ is $($ kilogram $)$ $($ metre $)$ $($ second ${)}^{-2}$.

The S.I unit of $R^2$ is $($ metre ${)}^2$.

$\therefore$ The S.I unit of $G{M}_e{M}_s$ OR $F{R}^2$ is $($ kilogram$)$$($ metre ${)}^3$$($ second ${)}^{-2}$

Also, Work done, $W=V\times q$ has the dimension of energy i.e $($ kilogram $)$ $($ metre ${)}^2$ $($ second ${)}^{-2}$

Hence, $($ Volt $)$$($ coulomb $)$$($ metre $)$ has the same dimension as that of $F{R}^2$.

(B) : Using, $v_{rms}=\sqrt{\frac{3RT}{M}}⟹\frac{3RT}{M}=v_{rms}^2$

$\therefore$ S.I unit of $\frac{3RT}{M}$ is $($ metre ${)}^2$$($ second ${)}^{-2}$

Also $C{V}^2$ has the dimension of Energy $\left(E\right)$ i.e $($ kilogram $)$$($ metre ${)}^2$$($ second ${)}^{-2}$

$\therefore$ $($ farad $)$$($ volt ${)}^2$$($ kilogram ${)}^{-1}$ represents $\frac{E}{M}$ i.e $($ metre ${)}^2$ $($ second ${)}^{-2}$

(C) : Using, $F=qvB⟹\frac{F^2}{q^2{B}^2}=v^2$

$\therefore \frac{F^2}{q^2{B}^2}$ has the dimension $($ metre ${)}^2$ $($ second ${)}^{-2}$

From (B), $\frac{F^2}{q^2{B}^2}$ has the same dimension as that of $($ farad $)$ $($ volt ${)}^2$ $($ kilogram ${)}^{-1}$

(D) : Using, $F=\frac{G{M}_{e}m}{R_e^2}⟹\frac{G{M}_e}{R_e}=\frac{F{R}_e}{m}$

As workdone, $W=FR$ has the dimension of energy i.e $($ kilogram $)$ $($ metre ${)}^2$ $($ second ${)}^{-2}$

$\therefore \frac{F{R}_e}{m}$ $=\frac{W}{m}$ which has the dimension $($ metre ${)}^2$ $($ second ${)}^{-2}$

From (B), $\frac{G{M}_e}{R_e}$ has the same dimension as that of $($ farad $)$ $($ volt ${)}^2$ $($ kilogram ${)}^{-1}$