Test if the following are dimensionally correct:

a) $h=\frac{2 \sec \theta}{\operatorname{prg}}$

Here, $h=[L], \mathrm{S}=\mathrm{F} / \mathrm{L}=\left[\mathrm{MT}^{-2}\right], \rho=\left[\mathrm{ML}^{-3}\right], \mathrm{r}=[\mathrm{L}], \mathrm{g}=\left[\mathrm{LT}^{-2}\right]$ $2 \operatorname{seos} \theta$

So, $\left.\overline{\rho r g}=\left[\mathrm{MT}^{-2}\right] / \mathrm{ML}^{-3} \mathrm{~L}^{2} \mathrm{~T}^{-2}\right]=[\mathrm{L}]$

This relation is correct

b) Here, $[v]=\left[L T^{-1}\right], P=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]$ and $\rho=\left[\mathrm{ML}^{-3}\right]$

Now, $\mathrm{P} / \rho=\left[L^{2} \mathrm{~T}^{-2}\right]$, so, $\sqrt{\frac{\rho}{\rho}}=\left[\mathrm{LT}^{-11}\right]=\mathrm{v}$.

This relation is correct

c) $\mathrm{V}=\frac{\frac{m P t r^{4}}{8 \eta{ }^{4}}}{}$

$\mathrm{V}=\left[\mathrm{L}^{3}\right], \mathrm{P}=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right], \mathrm{t}=[\mathrm{T}], \mathrm{r}=[\mathrm{L}],{ }^{\eta}=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-1}\right]$,

This relation is correct

d) $v=\left[\mathrm{T}^{-1}\right], m=[\mathrm{M}], \mathrm{g}=\left[\mathrm{LT}^{-2}\right], \mathrm{I}=\left[\mathrm{ML}^{2}\right]$ Now, mgl/l $\left.=\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right] / \mathrm{ML}^{2}\right]=\left[\mathrm{T}^{-2}\right]$.Now, ${ }^{\frac{1}{2 \pi} \sqrt{\frac{m g l}{I}}}=\left[\mathrm{T}^{-1}\right]$ Thus, this relation is also correct.