# Is the acceleration due to a fictitious force independent of mass in general?

Intuitively (at least to me) it seems that the answer should be "yes", since a fictitious force arises due to being in a non-inertial frame; the frame is accelerating, but the objects within this frame (neglecting any other forces) and so they will remain at rest (or at constant velocity), however, relative to an observer in this non-inertial frame, they will all seem to accelerate in the opposite direction at the same rate.

If the above is correct, then this notion can be shown mathematically as follows (at least, I think). Newton's 2nd law states that the acceleration, $\mathbf{a}$ of a massive object (due to an applied force) is proportional to the force, $\mathbf{F}$ acting on it, such that $\mathbf{F}=m\mathbf{a}$, where the mass $m$ of the object is the constant of proportionality. This does not define the force (by which I mean it doesn't give a mathematical expression describing the force, for example, the Coulomb force is defined by the mathematical expression $\mathbf{F}_{C}=\frac{1}{4\pi\varepsilon_{0}}\frac{q_{1}q_{2}}{r^{2}}\hat{\mathbf{r}}$ which is equal to $m\mathbf{a}$). However, in the case of a fictitious force, $\mathbf{F}_{fic}$, the force is defined by the mathematical expression $\mathbf{F}_{fic}=-m\mathbf{a}_{inert}$, where $\mathbf{a}_{inert}$ is the acceleration of the reference frame. It has this form since it is introduced to account for the acceleration of the reference frame. Then, by Newton's 2nd law, assuming that no other forces are acting, we have that $$-m\mathbf{a}_{inert}=m\mathbf{a}\Rightarrow\mathbf{a}=\mathbf{a}_{inert}$$ i.e. the acceleration of each of the objects due to this fictitious force is independent of its mass.

I'm unsure though whether I've conducted this analysis completely correctly though, so any feedback would be much appreciated.

• It seems like you're really asking whether fictitious forces are proportional to mass in general. Or, equivalently, whether acceleration due to fictitious forces is independent of mass. Apr 14, 2016 at 23:54
• @Brionius Yes, that is what I'd like to check my understanding of. Would what I put be correct at all? It seems that by definition the acceleration did to the fictitious force should be independent of mass since it is introduced to account for the fact that the reference frame itself is accelerating, and hence all objects within it are being accelerated at the same rate by virtue of being in this frame. Apr 15, 2016 at 7:38

• @user35305 actually that depends on situations like if you have centripetal force then all objects about a certain distance from axis would have the same acceleration as you know $a={\omega}^2R$ when you are on a train thats accelerating then all objects feel the same acceleration but at exactly opposite direction. Mar 6, 2019 at 0:38