The time dependence of a physical quantity $P$ is given by $P = P_0 e^{-\alpha t^2}$,where $\alpha$ is a constant and $t$ is the time. Then the constant $\alpha$ has:

The frequency $(v)$ of an oscillating liquid drop may depend upon the radius $(r)$ of the drop,the density $(\rho)$ of the liquid,and the surface tension $(s)$ of the liquid as: $v = r^{a} \rho^{b} s^{c}$. The values of $a, b,$ and $c$ respectively 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)

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 $L^{-1}$

Planck's constant $h$,speed of light $c$,and gravitational constant $G$ are used to form a unit of length $L$ and a unit of mass $M$. Then the correct option$(s)$ is(are):
$(A)$ $M \propto \sqrt{c}$
$(B)$ $M \propto \sqrt{G}$
$(C)$ $L \propto \sqrt{h}$
$(D)$ $L \propto \sqrt{G}$

The foundations of dimensional analysis were laid down by