If the capacitance of a nanocapacitor is measured in terms of a unit $u$ made by combining the electric charge $e,$ Bohr radius $a_0,$ Planck's constant $h$ and speed of light $c,$ then

The $SI$ unit of energy is $J = kg \, m^{2} \, s^{-2}$; that of speed $v$ is $m \, s^{-1}$ and of acceleration $a$ is $m \, s^{-2}$. Which of the formulae for kinetic energy $(K)$ given below can you rule out on the basis of dimensional arguments ($m$ stands for the mass of the body):
$(a)$ $K = m^{2} v^{3}$
$(b)$ $K = (1/2) m v^{2}$
$(c)$ $K = m a$
$(d)$ $K = (3/16) m v^{2}$
$(e)$ $K = (1/2) m v^{2} + m a$

If $L$,$C$,and $R$ denote the inductance,capacitance,and resistance respectively,the dimensional formula for $C^{2}LR$ is

Let us consider a system of units in which mass and angular momentum are dimensionless. If length has dimension of $L$,which of the following statement$(s)$ is/are correct?
$(1)$ The dimension of force is $L^{-3}$
$(2)$ The dimension of energy is $L^{-2}$
$(3)$ The dimension of power is $L^{-5}$
$(4)$ The dimension of linear momentum is $L^{-1}$

If $y$ represents pressure and $x$ represents velocity gradient,then the dimensions of $\frac{d^2 y}{d x^2}$ are

$A$ 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)