To find the distance $d$ over which a signal can be seen clearly in foggy conditions, a railways 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 find that $d$ is proportional to $S ^{1 / n}$. The value of $n$ is:

With the usual notations, the following equation ${S_t} = u + \frac{1}{2}a(2t - 1)$ is

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 $L ^{-1}$

The volume of a liquid flowing out per second of a pipe of length $l$ and radius $r$ is written by a student as $V\, = \,\frac{{\pi p{r^4}}}{{8\eta l}}$ where $p$ is the pressure difference between the two ends of the pipe and $\eta $ is coefficent of viscosity of the liquid having dimensional formula $[M^1L^{-1}T^{-1}] $. Check whether the equation is dimensionally correct.

A massive black hole of mass $m$ and radius $R$ is spinning with angular velocity $\omega$. The power $P$ radiated by it as gravitational waves is given by $P=G c^{-5} m^{x} R^{y} \omega^{z}$, where $c$ and $G$ are speed of light in free space and the universal gravitational constant, respectively. Then,

If force $(F)$, length $(L)  $ and time $(T)$ are assumed to be fundamental units, then the dimensional formula of the mass will be