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

Equation	Answer
$x = x_0 + v_0 t + (1/2)a t^2$	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 $[kT] = M\cdot L^2/T^2.$ Right side of the equation is also "dimensionless", since $(M\cdot L^2/T^2)(M\cdot L^2/T^2) = 1.$

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