Dimensional analysis

Dimensional analysis is a method that is used to check if an expression that has been derived could be incorrect. Suppose a problem involves a mass $m$ [kg], a length $L$ [m], a time $t$ [s], and a force $F$ [N]. You are asked to calculate a velocity. The expressions $3L/t$ and $\pi\frac{Ft}{m}$ could possibly be correct because they have the units of [m/s]. The expresions $3Lt$ and $\pi\frac{F}{m}$ must be incorrect because they do not have the units of [m/s].

Whenever you derive an expression, you should check the units. If the units are wrong, you have made a mistake.

The argument of a function such as $\sin$, $\cos$, $\exp$, of $\log$ must be unitless. An expression like $\sin\left(\frac{Ft^2}{mL}\right)$ could be correct while $\sin\left(\frac{Ft}{mL}\right)$ must be wrong.

The following variables are defined:

$x$, $L$, and $d$ are lengths and have units of [m]
$t$ and $\tau$ are times and have units of [s]
$F$ is a force and has units of [N]
$f$ is a frequency and has units of [Hz]
$\omega$ is an angular frequency and has units of [rad/s]
$v$ is a velocity and has units of [m/s]
$a$ is an acceleration and has units of [m/s²]
$E$ is an energy and has units of [J]
   $m$ is a mass and has units of [kg]
$T$ is the absolute temperature and has units of [K]
$V$ is a voltage and has units of [V]
$c$ is the speed of light and has units of [m/s]
$e$ is the electron charge and has units of [C]
$\hbar$ is Planck's constant and has units of [J s]
$k_B$ is Boltzmann's constant and has units of [J/K]

In a certain problem, must be calculated. Select the expressions which have the correct units. Any number of the expressions can be correct including all of them and none of them.

SI Units
Physical constants