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 $\left( ms ^{-1}\right)^{-1}$ respectively. The units of $y$ and $\gamma$ are

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

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 $\beta \gamma=\alpha \Rightarrow \gamma=\frac{ T ^{-1}}{ L ^{-1} T }$ $\Rightarrow \left[ LT ^{-2}\right]$
and $y=x^{2} \Rightarrow \left[L^{2}\right]$
$\alpha= s ^{-1} \Rightarrow \left[ T ^{-1}\right], \beta=\left[ LT ^{-1}\right]^{-1}$ $\Rightarrow \left[ L ^{-1} T \right]$
$y=m^{2}$