Question

In equation $y=x^2{\cos }^{2}2\pi \frac{\beta \gamma }{\alpha }$, the units of $x,\alpha ,\beta$ are $m,s^{-1}$ and ${\left(m{s}^{-1}\right)}^{-1}$ respectively. The units of $y$ and $\gamma$ are

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

Answer 1

Answer: (A)

$y=x^2{\cos }^{2}2\pi \left(\frac{\beta \gamma }{\alpha }\right)$
The argument of a trigonometric ratio is always dimensionless.
$\frac{\beta \gamma }{\alpha }=\left[M^0{L}^0{T}^0\right]$
or $\beta \gamma =\alpha \Rightarrow \gamma =\frac{T^{-1}}{L^{-1}T}$ $\Rightarrow \left[L{T}^{-2}\right]$
and $y=x^2\Rightarrow \left[L^2\right]$
$\alpha =s^{-1}\Rightarrow \left[T^{-1}\right],\beta ={\left[L{T}^{-1}\right]}^{-1}$ $\Rightarrow \left[L^{-1}T\right]$
$y=m^2$