DIMENSIONAL ANALYSIS AND ITS APPLICATIONS

If e is the electronic charge, c is the speed of light in free space and h is Planck’s constant, the quantity $\frac{1}{4\pi {\epsilon }_0}\frac{{\left|e\right|}^2}{hc}$ has dimensions of

If time (t), velocity $\left(\upsilon \right)$ <!-- [if gte mso 9]><xml> </xml><![endif]-->, and angular momentum $\left(\mathcal{l}\right)$ <!-- [if gte mso 9]><xml> </xml><![endif]--> are taken as the fundamental units. Then the dimension of mass (m) in terms of t, $\upsilon ,and\mathcal{l}$ <!-- [if gte mso 9]><xml> </xml><![endif]--> is:

If velocity [V], time [T] and force [F] are chosen as the base quantities, the dimensions of the mass will be:

Identify the pair of physical quantities that have same dimensions

Given below are two statements: One is labeled as Assertion (A) and other is labeler as Reason (R).

Assertion (A) : Time period of oscillation of a liquid drop depends on surface tension (S), if density of the liquid is <!-- [if gte mso 9]><xml> </xml><![endif]--> and radius of the drop is r, then

$T=K\sqrt[]{\raisebox{1ex}{$\rho r^3$}\!\left/ \!\raisebox{-1ex}{$S^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$}\right.}$  is dimensionally correct, where K is dimensionless.

Reason (R) : Using dimensional analysis we get R.H.S. having different dimension than that of time period.

In the light of above statements, choose the correct answer from the options given below.