Question

In equation $y=x^{2}co{s}^{2}2\pi \frac{\beta y^{\prime }}{\alpha }$, if the units of $x,\alpha ,\beta$ are $m,s^{-1}$and $({ms}^{-1}{)}^{-1}$ respectively, then units of y and y' are

• A. $m^2,{ms}^{-2}$
• B. $m,{ms}^{-1}$
• C. $m^2,m$
• D. $m,{ms}^{-2}$

Answer 1

Answer: (A)

$m^2,{ms}^{-2}$

Since argument of cos has to be dimensionless, thus $|2\pi \beta y^{\prime }/\alpha |=M^0{L}^0{T}^0$
Thus, $|y^{\prime }|=\frac{|\alpha |}{|\beta |}$ ($\pi$ is dimensionless)
Thus, $|y^{\prime }|=\frac{s^{-1}}{(m{s}^{-1}{)}^{-1}}=m{s}^{-2}$
Now, since cos is dimensionless, thus $|y|=|x^2|=m^2$