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.

A particle of mass $\mathrm{m}$ is tied to a string and swung around in a circular path of radius $\mathrm{r}$ with a constant speed $\mathrm{v}.$ Derive a formula for the centripetal force $\mathrm{F}$ exerted by the particle on our hand, using the method of dimensions.

The energy $\mathrm{E}$ of a particle oscillating in S.H.M depends on the mass $\mathrm{m}$ of the particle, frequency $\mathrm{n}$ and amplitude $\mathrm{a}$ of oscillation. Show dimensionally that $\mathrm{E}\propto \mathrm{m} {\mathrm{n}}^2 {\mathrm{a}}^2.$

The velocity of transverse waves along a string may depend upon the length $\mathrm{l}$ of the string, tension $\mathrm{f}$ in the string and mass per unit length $\mathrm{m}$ of the string. Derive a possible formula for the velocity dimensionally.

The frequency $\mathrm{n}$ of a tuning fork depends upon the length $l$ of the prong, the density $\mathrm{\rho }$ and the young's modulus $\mathrm{Y}$ of its material. From dimensional considerations, find a possible formula for the frequency of tuning fork.

The frequency $\mathrm{n}$ of an oscillating liquid drop may depend upon the radius $\mathrm{r}$ of the drop, density $\mathrm{\rho }$ and surface tension $S$ of the liquid. Obtain a formula for the frequency by the method of dimensions.