In dimensional consistency checks is $\mathrm{M}=\sqrt{3}\mathrm{M}$ true?

Question

When checking an equation with dimensional consistency, if I get
$\mathrm{M}=\sqrt{3}\mathrm{M}$

should I be worried? or the coefficients don't matter because we are concerned about dimensions?

Answer 1

Assuming the $\sqrt{3}$ is simply a non-dimensional constant, there is no problem

As a similar example the period of a basic pendulum is measured with the dimension of time. The obvious factors potentially affecting the period are mass, length (distance) and gravitational acceleration (distance divided by the square of time). Dimensional analysis of these would suggest the period could be proportional to $\sqrt{\frac{L}{g}}$, while physical analysis would suggest something close to simple harmonic motion with $$T \approx 2\pi \sqrt\frac{L}{g}$$ The $2\pi$ is a non-dimensional constant and neither affects the correctness of the dimensional analysis nor can be found by it