Using the method of dimensions, check the correctness of the following equations:

$v_e=\sqrt{\frac{2{\mathrm{GM}}_{\mathrm{e}}}{R_{\mathrm{e}}}}$, where $v_e$is the escape velocity from earth whose mass and radius are $M_e$and $R_e$ respectively.

Using the method of dimensions, check the correctness of the following equations:

$\mathrm{h}=\frac{3\mathrm{S} \mathrm{cos\theta }}{\mathrm{r\rho g}}$, where $h$is the height of a liquid of density $\rho$ and surface tension $S$ raised in a capillary tube of radius $r$, and $\theta$ is the angle of contact.

Using the method of dimensions, check the correctness of the following equations:

$V=\frac{\mathrm{\pi }p{r}^4}{8\mathrm{\eta l}}$, where $V$ is the volume of a liquid of coefficient of viscosity $\eta$ flowing per second through a horizontal capillary tube of radius $r$ and length $l$, having a pressure difference $p$ between its two ends.

Using the method of dimensions, check the correctness of the following equations:

$v=\sqrt{\frac{F}{m}}$, where $v$ is the velocity of a transverse wave in a string of mass per unit length $m$ and stretched under tension $F$.

Using the method of dimensions, check the correctness of the following equations:

$v=\sqrt{\frac{B}{\rho }}$, where $v$ is the velocity of a longitudinal wave in a liquid of bulk modulus of elasticity $B$ and density $\rho$.

Using the method of dimensions, check the correctness of the following equations:

$\lambda =\frac{h}{mv}$, where $\lambda$ is the de Broglie wavelength of a particle of $m$ moving with velocity $\nu$.

Prove with the help of dimensional analysis that the equation $h=\frac{1}{2}gt$ for the distance travelled by a body falling freely under gravity in time $t$ is incorrect. Find the correct equation with the help of dimensions.

Show dimensionally that the equation of the time-period of a simple pendulum of length $l$, is given by $\mathrm{t}=2\mathrm{\pi }\sqrt{\raisebox{1ex}{$\mathrm{l}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{g}$}\right.}$. .

Check the correctness of the relation $h=\frac{r\rho g}{2S}$ for the height of a liquid of density $\rho$ and surface tension $S$, raised in a capillary tube of radius $r$ and angle of contact zero with the liquid. If incorrect, then deduce the correct form.