A liquid drop placed on a horizontal plane has a near spherical shape (slightly flattened due to gravity). Let $R$ be the radius of its largest horizontal section. A small disturbance causes the drop to vibrate with frequency $v$ about its equilibrium shape. By dimensional analysis, the ratio $\frac{v}{\sqrt{\sigma / \rho R^3}}$ can be (Here, $\sigma$ is surface tension, $\rho$ is density, $g$ is acceleration due to gravity and $k$ is an arbitrary dimensionless constant)
$k \rho g R^2 / \sigma$
$k \rho R^3 / g_\sigma$
$k \rho R^2 / g \sigma$
$k \rho / g \sigma$
In the relation $y = a\cos (\omega t - kx)$, the dimensional formula for $k$ is
Two quantities $A$ and $B$ have different dimensions. Which mathematical operation given below is physically meaningful
A force $F$ is given by $F = at + b{t^2}$, where $t$ is time. What are the dimensions of $a$ and $b$
Let $[{\varepsilon _0}]$ denotes the dimensional formula of the permittivity of the vacuum and $[{\mu _0}]$ that of the permeability of the vacuum. If $M = {\rm{mass}}$, $L = {\rm{length}}$, $T = {\rm{Time}}$ and $I = {\rm{electric current}}$, then