If the buoyant force $F$ acting on an object depends on its volume $V$ immersed in a liquid,the density $\rho$ of the liquid,and the acceleration due to gravity $g$,then the correct expression for $F$ can be:

An artificial satellite is revolving around a planet of mass $M$ and radius $R$ in a circular orbit of radius $r$. From Kepler's third law about the period of a satellite around a common central body,the square of the period of revolution $T$ is proportional to the cube of the radius of the orbit $r$. Show using dimensional analysis that $T = \frac{k}{R}\sqrt{\frac{r^3}{g}}$,where $k$ is a dimensionless constant and $g$ is the acceleration due to gravity.

The density of a material in $SI$ units is $128 \ kg \ m^{-3}$. In a system of units where the unit of length is $25 \ cm$ and the unit of mass is $50 \ g$,the numerical value of the density of the material is:

$A$ quantity $f$ is given by $f = \sqrt{\frac{hc^5}{G}}$,where $c$ is the speed of light,$G$ is the universal gravitational constant,and $h$ is Planck's constant. The dimension of $f$ is that of:

If momentum $(P)$,area $(A)$,and time $(T)$ are taken to be fundamental quantities,then the dimensional formula for energy is:

If $A, B$ and $C$ are three different physical quantities with different dimensional formulae, then the combination which can never give a proper physical quantity is