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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:
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Verified Answer
The correct answer is:
$\alpha t / \beta$
From principle of homogeneity
$\begin{aligned}
& {[F]=\left[\alpha t^2\right]=[\beta t]} \\
& {[\alpha]=\frac{[F]}{\left[t^2\right]} \text { and }[\beta]=\frac{[F]}{[t]}} \\
& \therefore \quad[\alpha][t]=[\beta] \\
& \therefore \quad \frac{\alpha t}{\beta}=\text { dimensionless }
\end{aligned}$
$\begin{aligned}
& {[F]=\left[\alpha t^2\right]=[\beta t]} \\
& {[\alpha]=\frac{[F]}{\left[t^2\right]} \text { and }[\beta]=\frac{[F]}{[t]}} \\
& \therefore \quad[\alpha][t]=[\beta] \\
& \therefore \quad \frac{\alpha t}{\beta}=\text { dimensionless }
\end{aligned}$
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