Question

A physical quantity $x$ depends on quantities $y$ and $z$ as $x=Ay+Bsin\left(Cz\right)$, where $A,B$ and $C$ are constants.
Which of the following have same dimensions?

• A. $x$ and $AY$
• B. $x$ and $B$
• C. $C$ and $Z^{-1}$
• D. All of the above

Answer 1

The correct option is D All of the above

The principle of homogeneity states that the dimensions of all terms of an equation are the same. So,

$\left[X\right]=\left[AY\right]=\left[B\right]$
since $\left[Sin\right(Cz\left)\right]=1$

Thus $\left(a\right)$ and $\left(b\right)$ are correct

Also, $Cz$ is an angle, it has no dimensions.
So,
$\left[Cz\right]=1\Rightarrow \left[C\right]=\frac{1}{\left[z\right]}=[z{]}^{-1}$
$\left(c\right)$ is also correct.

Thus option $\left(d\right)$