- #1
Nirmal Padwal
- 41
- 2
- Homework Statement
- Solutions to physical problems often involve functions like
$$ cos u, sin u, e^u, ln u, · · · $$
where the argument ##u ## is a scalar that depends on physical quantities such as time,
frequency, mass, etc. Explain why u must be dimensionless (that is, independent of
the system of units used for mass, length, and time).
- Relevant Equations
- .
This is my approach:
These quantities namely mass, length, and time, are all additive in nature. ##2 m + 3m = 5 m ##. If the argument of the functions mentioned in the problem statement is not dimensionless, then mass, length or time do not remain additive in the image space of the functions. ## e^{2 m} + e^{3m} \neq e^{5m}##
Furthermore, if the arguments are dimensionless, the functions which were originally dimensionless have to be now themselves will have to be associated with physical quantities.
Is this reasoning valid?
These quantities namely mass, length, and time, are all additive in nature. ##2 m + 3m = 5 m ##. If the argument of the functions mentioned in the problem statement is not dimensionless, then mass, length or time do not remain additive in the image space of the functions. ## e^{2 m} + e^{3m} \neq e^{5m}##
Furthermore, if the arguments are dimensionless, the functions which were originally dimensionless have to be now themselves will have to be associated with physical quantities.
Is this reasoning valid?