Question

Suppose a quantity X can be dimensionally represented in terms of M, L and T that is, \[\left[ X \right] = {M^a}{L^b}{T^c}\]. The quantity mass\[\]

A. Can always be dimensionally represented in terms of L, T and X.
B. Can never be dimensionally represented in terms of L, T and X.
C. May be represented in terms of L, T and X if \[a = 0\].
D. May be represented in terms of L, T and X if \[a \ne 0\].

Answer 1

Hint: The power of any quantity if zero then the value of the given quantity becomes 1. Therefore, mass M can be represented in terms of L, T and X or not depending on the power of the quantity that is M itself. So, the value of mass M here is depended on its power that is a.





Formula used
\[\left[ X \right] = {M^a}{L^b}{T^c}\] (given in question itself)
Where, X is quantity.
M is mass.


Complete answer:


Given relation in the question:
\[\left[ X \right] = {M^a}{L^b}{T^c}\]
Here X is represented in the terms of M, L and T.
It is a condition for representing mass M in terms of X, L and T.
Condition: It depends on the power of mass M that is ‘a’ here.
If \[a = 0\]: then the mass M cannot be represented in terms of X, L and T;
As the value of \[{M^a} = {M^0} = 1\] , so there will be no relation at all.
If \[a \ne 0\]: then the mass M can be represented in terms of X, L and T;
As the value of \[{M^a}\] will not become 1. So, there will be a relationship between them.
Therefore, for option A and B; it is wrong as always and never condition cannot be applied.
Always cannot be represented is the wrong option as for a = 0, it cannot be represented.
Never cannot be represented as the wrong option as for \[a \ne 0\], it can be represented.

Hence, the correct answer is Option D.





Note: Only when the power of any quantity is zero then the value of quantity becomes 1. For any other value of power mass M can be represented in terms of L, T and X or not depending on the power of the quantity that is M itself. Also the power of each quantity matters too.