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:

Column I		Column II
A	$G{M}_e{M}_s$ $G$ - universal gravitational constant, $M_e$ - mass of the Earth, $M_s$ - mass of the Sun.	P	(volt)(coulomb) (metre)
B	$\frac{3RT}{M}$ $R$ - universal gas constant, $T$ - absolute temperature, $M$ - molar mass.	Q	$($ kilogram $)(\text{ metre }{)}^3(\text{ second }{)}^{-2}$
C	$\frac{F^2}{q^2{B}^2}$ $F-$ force, $q-$ charge, $B-$ magnetic field.	R	(metre) ${}^2(\text{ second }{)}^{-2}$
D	$\frac{G{M}_e}{R_e}$ $G$ - universal gravitational constant, $M_e$ - mass of the Earth, $R_e-$ radius of the Earth.	S	$($ farad $)(\text{ volt }{)}^2(kg{)}^{-1}$

• A. $(A)\to (P),(Q);(B)\to (R),(S);(C)\to (P),(S);(D)\to (R)$
• B. $(A)\to (P),(Q);(B)\to (R),(S);(C)\to (R),(S);(D)\to (R),(S)$
• C. $(A)\to (P),(Q);(B)\to (R),(S);(C)\to (R),(S);(D)\to (R),(S)$
• D. $(A)\to (P),(Q);(B)\to (R),(S);(C)\to (R),(S);(D)\to (R),(q)$

Answer 1

Answer: (B)

$(A)\to (P),(Q)$
Quantity given in option $(A)$ has a dimension of force $\times (\text{ length }{)}^2$;
quantities given in options $(P)$ and $(Q)$ has the same dimensions:
$G{M}_e{M}_s=F{r}^2$ or its unit is $kg{m}^3{s}^{-2}$.
(B) $\to (R),(S)$
We have $\frac{3RT}{M}=C_{\max }^2\Rightarrow$ the unit $m^2{s}^{-2}$,
which is nothing but energy per unit mass; the dimensions of this quantity is $L/T^2$;
quantities given in options $(R)$ and $(S)$ have the same dimension.
$(C)\to (R),(S)$
We have $\frac{F^2}{q^2{B}^2}=v^2\Rightarrow$ the unit $m^2{s}^{-2}$,
that is, the condition is same as that of option (B).
$(D)\to (R),(S)$
It is square of the escape velocity of the Earth:
$\frac{GMe}{Re}=V_g=\frac{W}{m}\Rightarrow$ the unit $;m^2{s}^{-2}$
The dimensions are same as that of option (B).