A force defined by $F = \alpha t^2 + \beta t$ acts on a particle at a given time $t$. The factor which is dimensionless, if $\alpha$ and $\beta$ are constants, is:

To identify the dimensionless factor, we apply the principle of homogeneity of dimensions, which states that every term in a physical equation must possess the same dimensions .

1. Determine dimensions of constants: In the equation $F = \alpha t^2 + \beta t$, the dimensions of force ($F$) are $[M L T^{-2}]$ .

• For the term $\alpha t^2$: $[\alpha][T^2] = [M L T^{-2}] \Rightarrow [\alpha] = [M L T^{-4}]$.
• For the term $\beta t$: $[\beta][T] = [M L T^{-2}] \Rightarrow [\beta] = [M L T^{-3}]$.

2. Test the options:

• Option A ($\alpha t / \beta$): Dimensionality is $([M L T^{-4}] \cdot [T]) / [M L T^{-3}] = [M L T^{-3}] / [M L T^{-3}] = [M^0 L^0 T^0]$. This factor is dimensionless.
• Option D ($\beta t / \alpha$): Dimensionality is $([M L T^{-3}] \cdot [T]) / [M L T^{-4}] = [M L T^{-2}] / [M L T^{-4}] = [T^2]$. This is not dimensionless.

Therefore, the factor $\alpha t / \beta$ has no dimensions.