Question

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

• A. $x$ and $B$
• B. $C$ and $z^{-1}$
• C. $y$ and $B/A$
• D. $x$ and $A$

Answer 1

Answer: (D)

Step 1: Given that:

A physical quantity x depends on y and z as;

$x=Ay+B\tan \left(Cz\right)$

Step 2: Comparing the dimension of quantities of the given relation:

According to the principle of homogeneity, both sides will have the same dimensions for each term.

Therefore, we have,

The dimension of x = Dimension of Ay = Dimension of B

(As the angular part that is Cz will be dimensionless.)

Thus,

[x] = [Ay] = [B]

[$\frac{x}{A}$ ] = [y] = [$\frac{B}{A}$ ]

Thus,

The dimension of x and A is not the same.

Hence,

Option B) x and A is the correct option.