In a practical unit system,if the unit of mass becomes double and that of time becomes half,then $8 \text{ joule}$ will be equal to .............. unit of work.

If the gravitational constant $(G)$,Planck's constant $(h)$,and the velocity of light $(c)$ are chosen as fundamental units,the dimension of the radius of gyration is:

State the principle of homogeneity of dimensions.

To find the distance $d$ over which a signal can be seen clearly in foggy conditions,a railway engineer uses dimensional analysis and assumes that the distance depends on the mass density $\rho$ of the fog,intensity (power/area) $S$ of the light from the signal,and its frequency $f$. The engineer finds that $d$ is proportional to $S^{1/n}$. The value of $n$ is:

Consider the equation $H = \frac{x^p \epsilon^q E^r}{t^s}$, where $H = \text{magnetic field intensity}$, $E = \text{electric field}$, $\epsilon = \text{permittivity}$, $x = \text{distance}$, and $t = \text{time}$. The values of $p, q, r$, and $s$ respectively are:

If velocity of light $c$,Planck's constant $h$,and gravitational constant $G$ are taken as fundamental quantities,then express time in terms of dimensions of these quantities.