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 \((M\cdot L^2/T^2)(M\cdot L^2/T^2) = 1.\)