Question
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:
Options
Solution
Sol. [Key Concept: Impulse = ∫F dt for a time-varying force]
From principle of homogeneity
$$\begin{matrix} & \lbrack F\rbrack = \left\lbrack \alpha t^{2} \right\rbrack = \lbrack\beta t\rbrack \\ & \lbrack\alpha\rbrack = \frac{\lbrack F\rbrack}{\left\lbrack t^{2} \right\rbrack}\text{~}\text{and}\text{~}\lbrack\beta\rbrack = \frac{\lbrack F\rbrack}{\lbrack t\rbrack} \end{matrix}$$
$$\begin{matrix} \therefore & \lbrack\alpha\rbrack\lbrack t\rbrack = \lbrack\beta\rbrack \\ \therefore & \frac{\alpha t}{\beta} = \text{~}\text{dimensionless}\text{~} \end{matrix}$$
Create a free account to view solution
View Solution Free