Question

Suppose a quantity x can be dimensionally represented in terms of M, L and T, that is, $[x] = M^a L^b T^c$. The quantity mass

• A. can always be dimensionally represented in terms of L, T and x,
• B. can never be dimensoinally represented in terms of

Answer 1

Neither option (a) ("always") nor (b) ("never") is correct for all cases since mass can be represented in terms of L , T , and x if and only if a != 0

Answer 2

Let the dimensional formula of $x$ be

$$[x] = M^a L^b T^c.$$

Suppose we want to represent mass $M$ in terms of $x$, $L$, and $T$. Assume

$$M = x^\alpha L^\beta T^\gamma.$$

Taking dimensions on both sides, we have

$$M^1 = (M^a L^b T^c)^\alpha L^\beta T^\gamma = M^{a\alpha} L^{b\alpha+\beta} T^{c\alpha+\gamma}.$$

For dimensional homogeneity, we require:

$$\begin{cases} a\alpha=1,\\[1mm] b\alpha+\beta=0,\\[1mm] c\alpha+\gamma=0. \end{cases}$$

From the first equation, we obtain

$$\alpha = \frac{1}{a}.$$

This representation is possible only if $a \neq 0$. If $a=0$, then $x$ has no dependency on mass, and it is impossible to construct the dimension of mass using $x, L$ and $T$.

Thus, mass can be represented in terms of $L$, $T$, and $x$ if and only if $a\ne 0$. In other words, the statement “mass can always be represented in terms of $L$, $T$ and $x$” is not true, and neither is the absolute statement “mass can never be represented…” since it depends on the value of $a$.