An object is moving through the liquid. The viscous damping force acting on it is proportional to the velocity. Then dimension of constant of proportionality is

The foundations of dimensional analysis were laid down by

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 force of interaction between two atoms is given by $F\, = \,\alpha \beta \,\exp \,\left( { - \frac{{{x^2}}}{{\alpha kt}}} \right);$ where $x$ is the distance, $k$ is the Boltzmann constant and $T$ is temperature and $\alpha $ and $\beta $ are two constants. The dimension of $\beta $ is

$A, B, C$ and $D$ are four different physical quantities having different dimensions. None of them is dimensionless. But we know that the equation $AD = C\, ln\, (BD)$ holds true. Then which of the combination is not a meaningful quantity ?

The potential energy of a particle varies with distance $x$ from a fixed origin as $U=\frac{A \sqrt{x}}{x^2+B}$, where $A$ and $B$ are dimensional constants then dimensional formula for $A B$ is