# How does damping constant relate to mass?

(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.

• Look here. physics.stackexchange.com/q/8495
– Dan
Feb 19, 2022 at 18:57
• 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. Feb 19, 2022 at 19:08

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.