The force F acting on a particle is given by the equation F = (α -...

According to the principle of dimensional homogeneity, quantities added or subtracted must have the same dimensions.

Therefore, the dimensions of $\alpha$ must be the same as those of $x^2$.
$[\alpha] = [x^2] = [\mathrm{L}^2]$

The dimensional formula for the entire equation is:
$[F] = \dfrac{[\alpha]}{[\beta][t]}$

Rearranging to find the dimensions of $\beta$:
$[\beta] = \dfrac{[\alpha]}{[F][t]} = \dfrac{[\mathrm{L}^2]}{[\mathrm{M} \mathrm{L} \mathrm{T}^{-2}][\mathrm{T}]}$
$[\beta] = \dfrac{[\mathrm{L}^2]}{[\mathrm{M} \mathrm{L} \mathrm{T}^{-1}]} = [\mathrm{M}^{-1} \mathrm{L} \mathrm{T}]$

Now, the dimensional formula of the ratio $\dfrac{\alpha}{\beta}$ is:
\left[\dfrac{\alpha}{\beta}\right] = \dfrac{[\mathrm{L}^2]}{[\mathrm{M}^{-1} \mathrm{L} \mathrm{T}]} = [\mathrm{M} \mathrm{L} \mathrm{T}^{-1}]$<br><br>Answer:$[\mathrm{M} \mathrm{L} \mathrm{T}^{-1}]