$A$ force defined by $F = \alpha t^2 + \beta t$ acts on a particle at a given time $t$. The factor which is dimensionless,if $\alpha$ and $\beta$ are constants,is:

What is Dimensional Analysis? State the uses of Dimensional Analysis.

Young-Laplace law states that the excess pressure inside a soap bubble of radius $R$ is given by $\Delta P = 4 \sigma / R$,where $\sigma$ is the coefficient of surface tension of the soap. The $EOTVOS$ number $E_0$ is a dimensionless number that is used to describe the shape of bubbles rising through a surrounding fluid. It is a combination of $g$ (acceleration due to gravity),$\rho$ (density of the surrounding fluid),$\sigma$ (surface tension),and a characteristic length scale $L$ (radius of the bubble). $A$ possible expression for $E_0$ is:

The force $F$ is given in terms of time $t$ and displacement $x$ by the equation $F = A \cos(Bx) + C \sin(Dt).$ The dimensional formula of $D/B$ is

If the speed of light $(c)$,acceleration due to gravity $(g)$,and pressure $(p)$ are taken as the fundamental quantities,then the dimension of the gravitational constant $(G)$ is:

The period of a body under $SHM$ is presented by $T = P^a D^b S^c$; where $P$ is pressure,$D$ is density,and $S$ is surface tension. The values of $a, b,$ and $c$ are: