Question

A physical quantity $x$ depends on quantities $y$ and $z$ as follows: $x = Ay + B\tan Cz$, where $A,B$ and $C$ are constants. Which of the following do not have the same dimensions?

• A. (a) $x$ and $B$
• A. (b) $C$ and $z^{- 1}$
• A. (c) $y$ and $B/A$
• A. (d) $x$ and $A$

Answer 1

(d)

Sol. $x = Ay + B\tan Cz$

From the dimensional homogeneity

$\lbrack x\rbrack = \lbrack Ay\rbrack = \lbrack B\rbrack \Rightarrow \left\lbrack \frac{x}{A} \right\rbrack = \lbrack y\rbrack = \left\lbrack \frac{B}{A} \right\rbrack$

$\lbrack Cz\rbrack = \lbrack M^{0}L^{0}T^{0}\rbrack =$ Dimension less

$x$ and $B$; $C$ and $Z^{- 1};y$ and $\frac{B}{A}$ have the same dimension but $x$ and $A$ have the different dimensions.