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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:
A
$\alpha\beta t$
B
$\frac{\alpha\beta}{t}$
C
$\frac{\beta t}{\alpha}$
D
$\frac{\alpha t}{\beta}$
Detailed Solution
$F = \alpha t^2 + \beta t$
$[\alpha t^2] = [M^1L^1T^{-2}] \Rightarrow [\alpha] = [M^1L^1T^{-4}]$
$[\beta t] = [M^1L^1T^{-2}] \Rightarrow [\beta] = [M^1L^1T^{-3}]$
$\left[\frac{\alpha t}{\beta}\right] = \frac{[M^1L^1T^{-4}][T]}{[M^1L^1T^{-3}]} = [M^0L^0T^0]$
So $\frac{\alpha t}{\beta}$ is dimensionless.