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(Moderator note: this question is not answered by a different post here)

In damped harmonic motion, I'm led to believe that the equation of motion in a mass-spring system is as follows $$x = Ae^{-λt} cos(wt)$$

After researching, I couldn't find a clear - explicit - relationship between λ and the mass of the object. I am aware of the identity relating the natural frequency to λ, but the frequency is influenced by λ itself.

Some people say that λ is proportional to the square root of mass, and some say that it is proportional to 1/sqrt(mass). What is the correct relationship between mass and λ in damped harmonic motion? Is it a power relationship, a linear relationship, a square root relationship... etc? Please do tell me why as well.

P.S. If you had to, please keep the calculus to a minimum. I'm only a senior high school student.

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  • $\begingroup$ Look here. physics.stackexchange.com/q/8495 $\endgroup$
    – Dan
    Commented Feb 19, 2022 at 18:57
  • $\begingroup$ A really useful post, and in fact my confusion stemmed from the position-time function featured in the answer post. Since $\lambda = b/2m$, does that mean that $\lambda$ is inversely proportional to $m$? Is $b$ a constant or does it change? Thanks. $\endgroup$
    – Alvin K.
    Commented Feb 19, 2022 at 19:08

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Questions that say essentially "Some people report A, whereas other people report B" without identifying the sources are difficult to address because (1) those people may be flat wrong or (2) they may be right for certain scenarios (but not this one) or (3) they may be right but being misinterpreted, etc.

Putting that aside, if I do a quick online search for underdamped oscillation equation, the first three sources that come up (1, 2, 3) are all consistent and unambiguous: $\lambda=\frac{c}{2m}$, where $c$ is the (constant) damping factor, i.e., the coupling coefficient between the speed of mass $m$ and the corresponding damping resistance in units of force. So $\lambda \sim\frac{1}{m}$, not $\frac{1}{\sqrt{m}}$ or $\sqrt{m}$.

A dimensional analysis provides further reassurance: $c$ has units of [force]/([distance]/[time])=[mass]/[time], and $m$ has units of [mass], so $\lambda$ has units of 1/[time], which combines with $t$'s units of [time] to correctly provide a nondimensional argument in the exponential function.

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