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

Answer 1

Answer: (D)

(D)

Answer 2

The given equation is $x = Ay + B \tan Cz$. According to the principle of dimensional homogeneity, the dimensions of each term on the right side must be equal to the dimension of the left side. Therefore, $[x] = [Ay]$ and $[x] = [B \tan Cz]$.

For the term $B \tan Cz$, the argument of a trigonometric function must be dimensionless. So, $[Cz] = 1$, which implies $[C] = [z]^{-1}$. Also, $[\tan Cz] = 1$. Therefore, $[B \tan Cz] = [B]$, and $[x] = [B]$.

For the term $Ay$, $[x] = [Ay]$, so $[A] = \frac{[x]}{[y]}$.

Now let's check the dimensions of the pairs given in the options:

(A) $x$ and $B$: Since $[x] = [B]$, they have the same dimensions.

(B) $C$ and $z^{-1}$: Since $[C] = [z]^{-1}$, they have the same dimensions.

(C) $y$ and $B/A$: $[\frac{B}{A}] = \frac{[B]}{[A]} = \frac{[x]}{[x]/[y]} = [y]$. So, $y$ and $B/A$ have the same dimensions.

(D) $x$ and $A$: $[A] = \frac{[x]}{[y]}$. For $x$ and $A$ to have the same dimensions, we must have $[x] = [A]$. Substituting the expression for $[A]$, we get $[x] = \frac{[x]}{[y]}$. This equation holds only if $[y] = 1$ (i.e., $y$ is a dimensionless quantity). The problem does not state that $y$ is dimensionless. Therefore, $x$ and $A$ do not have the same dimensions, assuming $y$ is a quantity with dimensions.

Thus, the answer is (D).