# 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 \(t\) |

\(P = \sqrt {\rho g h}\) |
where \(P\) is pressure |

\(1n N_d/N_a = - [Vgh_d (\rho - \rho_1)] kT\) |
where \(N_d\) and \(N_a\) are number of particles |

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