A force defined by F = αt^2 + βt acts on a particle at a given time t. The factor which is dimensionless,

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:

(1) \(\frac{\beta t}{\alpha}\)

(2) \(\frac{\alpha t}{\beta}\) 

(3) \(\alpha \beta t\) 

(4) \(\frac{\alpha \beta}{t}\) 

Answer 1

Correct option is : (2) \(\frac{\alpha t}{\beta}\) 

From principle of homogeneity 

\([F] = [\alpha t^2] = [\beta t]\) 

\([\alpha] = \frac{[F]}{[t^2]} \ and \ [\beta ] = \frac{[F]}{[t]}\) 

\(\therefore\ [\alpha]\ [t] = [\beta]\) 

\( \therefore\ \frac{\alpha t}{\beta}= \text{dimensionless}\)