Let $x = \pi R \left( \frac{P^2 - Q^2}{2} \right)$,where $P, Q$ and $R$ are lengths. The physical quantity $x$ is

The dimensional formula of a physical quantity represented by $\frac{e^2}{4 \pi \varepsilon_0 \hbar}$ is (where $e$ is the charge of an electron,$\varepsilon_0$ is the permittivity of free space,and $\hbar$ is the reduced Planck's constant). Note: The expression $\frac{e^2}{4 \pi \varepsilon_0 \hbar}$ is equivalent to the fine-structure constant $\alpha$ multiplied by the speed of light $c$.

Force $F = P \cos(Ax) + Q \sin(Bt)$,where $x$ and $t$ are displacement and time respectively. Which one of the following physical quantities has the dimensions of $\left[\frac{B}{A}\right]$?

What are the dimensions of the change in velocity?

The dimensions of $(\mu_0 \varepsilon_0)^{-1/2}$ are

The dimensional formula of the effective torsional constant of a spring is . . . . . . .