Question

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

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

Answer 1

The correct option is A ${\text{m}}^2,{\text{ms}}^{-2}$

In equation
$y=x^2{\cos }^{2}2\pi \left(\frac{\beta \gamma }{\alpha }\right)$
The term $\left(\frac{\beta \gamma }{\alpha }\right)$ is dimensionless.
$\Rightarrow \left[\frac{\beta \gamma }{\alpha }\right]=\left[M^0{L}^0{T}^0\right]$
or, $\left[\beta \gamma \right]=\left[\alpha \right]$
$\Rightarrow \left[\gamma \right]=\frac{\left[\alpha \right]}{\left[\beta \right]}=\frac{T^{-1}}{L^{-1}{T}^{+1}}$
Since, unit of $\alpha \to s^{-1}$ and $\beta \to (m{s}^{-1}{)}^{-1}$
$\Rightarrow \left[\gamma \right]=\left[L{T}^{-2}\right]$
$\Rightarrow$ Unit of $\gamma$ is $m{s}^{-2}$
Also, $\left[y\right]=\left[x^2\right]=\left[L^2\right]$
Therefore, units of $y$ and $\gamma$ are $m^2,m{s}^{-2}$ respectively.