# How can we write $F = ma$ if Force is frame-independent and acceleration is frame-dependent?

As we know acceleration is frame dependent and force is frame independent. In Newton's Second Law of Motion, how can we write $$F = ma$$? Doesn't that imply Frame independent = frame dependent?

Newton's laws of motion hold only in inertial frames of references.

In Newtonian mechanics, all of the inertial frames measure the same acceleration (because they don't accelerate one relative to the other).

• Thanks for the answer Oct 12 '21 at 2:38
• Yes and I will add a P.S.: in special relativity both the force and the acceleration can depend on inertial frame. One still has ${\bf f} = d{\bf p}/dt$ but in general now ${\bf f} \ne m {\bf a}$. I think this issue was not what the OP had in mind but I thought I would add it in for the sake of completeness. Oct 12 '21 at 16:48
• We are considering classical mechanics here so we can consider mass is constant Oct 14 '21 at 5:17

For inertial frames of references, $$F=ma$$ holds true. For non-inertial frames, to balance this equation with the new relative acceleration $$a'$$, you need to add pseudo forces, as in, $$F= ma' + F'$$. An example of this pseudo force would be the centrifugal force, which comes into the equation when you observe things from a rotating frame of reference.

• Really thankful for the answer Oct 12 '21 at 2:39

"Frames" are more of relativistic concept than a Newtonian one. In Newtonian physics, there's One True Frame. If we add Galilean boosts to Newtonian physics, acceleration and force are both frame independent. In relativity, both acceleration and force are frame dependent. In both of them, $$F=ma$$ (if $$m$$ refers to relativistic mass).