Question

Test if the following equations are dimensionally correct
(a) h=2S cosθrhorg,
(b) v=√Prho,
(c) V=π P r4t8 η l
(d) v=12 π√mglI;
where h = height, S = surface tension, rho = density, P = pressure, V = volume, η= coefficient of viscosity, v = frequency and I = moment of interia.

Answer 1

Following equations are tested and they are dimensionally correct.

Explanation:

$(a) h=\frac{2 s \cos \theta}{\rho r g}$

Dimensions of h $=[L]$

Dimensions of S $=\left[\mathrm{MT}^{-2}\right]$

Dimensions of ρ = $\left[\mathrm{ML}^{-3}\right]$

Dimensions of r = [L]

Dimensions of g = [L$T^-^2$]

Checking the R.H.S

Dimensionally ,

                           $=\frac{M T^{-2}}{\left[M L^{-3}\right][L]\left[L T^{-2}\right]}$

                           $=\left[\mathrm{M}^{0} \mathrm{L}^{1} \mathrm{T}^{0}\right]$ ]  is dimensionally correct

$(b) w=\sqrt{\frac{p}{\rho}}$

Dimensions of v = [L$T^-^1$]

Dimensions of P = $\left[\mathrm{ML}^{-1} \mathrm{T}^{-2}\right]$

Dimensions of ρ = $\left[\mathrm{ML}^{-3}\right]$

Checking the R.H.S

Dimensionally,

                         $=\sqrt{\frac{\left[M L^{-1} T^{-2}\right]}{\left[M L^{-3}\right]}}$

                         $=\sqrt{L^{2} T^{-2}}$

                         $=L T^{-2}$ is dimensionally correct

$(c) \ {v}=\frac{\pi P r^{4} t}{8 \eta 1}$

Dimensions of V = $\left[L^{3}\right]$

Dimensions of P = $\left[\mathrm{ML}^{-1} \mathrm{T}^{-2}\right]$

Dimensions of r4 = $\left[L^{4}\right]$

Dimensions of t = [T]

Dimensions of l = [L]

Dimensions of η = $\left[\mathrm{ML}^{-1} \mathrm{T}^{-1}\right]$

Checking the R.H.S

Dimensionally ,

                      $=\frac{\left[M L^{-1} T^{-2}\right]\left[L^{4}\right][T]}{\left[M T^{-1} L^{-1}\right][L]}$

                      $=\left[L^{3}\right]$ is dimensionally correct

$(d) V=\frac{1}{2 \pi} \sqrt{\frac{m g l}{l}}$

Dimensions of v = $\left[T^{-1}\right]$

Dimensions of m = [M]

Dimensions of g = $\left[L T^{-2}\right]$

Dimensions of l = [L]

Dimensions of I = [ML²]

Checking the R.H.S

Dimensionally ,

                         $=\sqrt{\frac{[M]\left[L T^{-2}\right][L]}{M L^{2}}}$

                         $=\left[T^{-1}\right]$ is dimensionally correct

Thus, the following equations are dimensinally correct.

Answer 2

Following equations are tested and they are dimensionally correct.

Explanation:

(a) h=\frac{2 s \cos \theta}{\rho r g}

Dimensions of h

=[L]

Dimensions of S

=\left[\mathrm{MT}^{-2}\right]

Dimensions of ρ =

\left[\mathrm{ML}^{-3}\right]

Dimensions of r = [L]

Dimensions of g = [L

?f=%24%24T%5E-%5E2%24%24

Checking the R.H.S

Dimensionally ,

=\frac{M T^{-2}}{\left[M L^{-3}\right][L]\left[L T^{-2}\right]}

=\left[\mathrm{M}^{0} \mathrm{L}^{1} \mathrm{T}^{0}\right]

]  is dimensionally correct

(b) w=\sqrt{\frac{p}{\rho}}

Dimensions of v = [L

?f=%24%24T%5E-%5E1%24%24

Dimensions of P =

\left[\mathrm{ML}^{-1} \mathrm{T}^{-2}\right]

Dimensions of ρ =

\left[\mathrm{ML}^{-3}\right]

ANSWER

Checking the R.H.S

Dimensionally,