Question

Explain this statement clearly:
“To call a dimensionless quantity ‘large’ or ‘small’ is meaningless without specifying a standard for comparison,” In view of this, which of the following statements are complete:
(a) atoms are very small objects
(b) a jet plane moves with great speed
(c) the mass of Jupiter is very large
(d) the air inside this room contains a large number of molecules
(e) a proton is much more massive than an electron
(f) the speed of sound is much smaller than the speed of light.

Answer 1

Hint: Physical quantities are described as large or small depending upon the standard or the unit of measurement. A dimensionless quantity is a quantity which has no physical dimensions, in other words it is independent of the fundamental units.

Complete answer:
The given statement is correct. Now, for an example we know that angle is a dimensionless quantity.
If we say that $\angle \theta = {70^ \circ }$ is greater than $\angle \theta = {20^ \circ }$, this statement is said to be meaningful because a standard is specified for comparison for the dimensionless quantity.
Now, from the given statements, statement (e) and statement (f) are correct because a standard is specified for the comparison. Even if the given physical quantity is dimensionless, we can find out whether it is small or large. While in the statements (a), (b), (c) and (d) a standard for comparison is needed to be specified to make it a meaningful statement.
Hence, in view of this, statements (e) and (f) are complete.

Additional Information:
Dimensional formula of a given physical quantity is defined as the expression which shows how and which of the fundamental quantities represent the dimensions of a physical quantity. The equation which is obtained by equating a physical quantity with its dimensional formula is called the dimensional equation of the given physical quantity.

Note:
Students tend to confuse between Dimensionless physical quantity and a Dimensional constant physical quantity. A dimensionless physical quantity is one which has dimensions. In other words, it would have no units associated with it. On the other hand, dimensional constants may or may not have units associated with it but its value is fixed and depends only on the chosen fundamental units being used to represent it, not the system itself.