Dimensional Analysis Quiz 7

Determine if the following equations are dimensionally correct.

Equation Dimensions
$x = x_0 + v_0 t + (1/2)a t^2$

where $x$ is the displacement at time $t$
$x_0$ is the displacement at time $t = 0$
$v_0$ is the velocity at time $t = 0$
$a$ is the constant acceleration

$P = \sqrt {\rho g h}$

where $P$ is pressure
$\rho$ is density
$g$ is gravitational acceleration
$h$ is height

$1n N_d/N_a = - [Vgh_d (\rho - \rho_1)] kT$

where $N_d$ and $N_a$ are number of particles
$V$ is volume
$g$ is gravitational acceleration
$h_d$ is distance
$\rho$ and $\rho_1$ are densitites
$k$ is Boltzmann's constant with SI units of joules per kelvin
$T$ is absolute temperature

$x = x_0 + v_0 t + (1/2)a t^2$ Dimensionally correct. Each term has dimensions of $L$.
$P = \sqrt {\rho g h}$ Not dimensionally correct.
${P} = M\cdot L^{-1} \cdot T^{-2}$
$\sqrt {\rho g h} = M^{1/2} \cdot L^{-1/2} \cdot T^{-1}$
$1n N_d/N_a = - [Vgh_d (\rho - \rho_1)] kT$ Dimensionally correct.  Left side of the equation is "dimensionless".
$[Vgh_d(\rho-\rho_1)] = M\cdot L^2/T^2.$
$kT$ has SI units of joules, (which is a unit  of energy), and therefore $[kT] = M\cdot L^2/T^2.$
since $(M\cdot L^2/T^2)(M\cdot L^2/T^2) = 1.$