This question has been bugging me for quite some time, I have seen some explanations which are mathematical and don't make sense to me, most of them talk about Galilean relativity, but I am looking for a nice simple explanation which can help me understand why $F$ is invariant since both, Mass $M$ and Acceleration $A$ are not invariant?
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1$\begingroup$ Why do you think mass and acceleration are not invariant? Are you asking about Galilean (Newtonian) relativity or special relativity? $\endgroup$– PukCommented Feb 20 at 21:01
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$\begingroup$ All inertial frames have the same proper acceleration of 0, and the rest mass is invariant in all frames. There is no case where rest mass and proper acceleration vary between inertial frames. If you require imaginary forces with no real physical cause to explain your observations, you are not in an inertial frame. $\endgroup$– Nuclear HoagieCommented Feb 20 at 21:05
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$\begingroup$ @NuclearHoagie Thank you, I understand it now, and yes Mass $M$ does not vary as far as inertial frames are concerned. $\endgroup$– bobby76Commented Feb 21 at 16:56
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$\begingroup$ @Puk Yes, that was a mistake I should have not said that, indeed mass $M$ stays constant in inertial frames. $\endgroup$– bobby76Commented Feb 21 at 16:57
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2 Answers
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I am looking for a nice simple explanantion which can help me understand why $F$ is invariant
How about a simple example:
If I apply a force $F$ to your arm while we are in a car, do you think it will feel different if we are accelerating together in the car (a non-inertial reference frame) than if the car were moving at constant velocity (an inertial frame)?
If the answer is no, $F$ is reference frame independent.
Hope this helps.
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$\begingroup$ Thanks, this is was the best explanation! $\endgroup$– bobby76Commented Feb 21 at 16:55
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Using newton's laws only, we have $F=ma$, inertia, and if $A$ exerts a force on $B$ then $B$ will exert an equal and opposite force on $A$
In a reference frame, call it $S$, we have a force $F = ma = m\frac{dv}{dt}$ which acts on you (let's say you're at the origin of $S$). If we applied a velocity boost $v'$ to $S$, we have instead of $F = ma = m\frac{dv}{dt}$: $F' = m\frac{d(v+v')}{dt}=m\frac{dv+dv'}{dt}$ since $v'$ is constant, this resolves to $m\frac{dv}{dt}$ since constant velocity does not change with time, which leaves you with the same force as in the non boosted frame. This is what is meant by $F=ma$ behaving the same in all inertial (non-accelerating) reference frames.
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$\begingroup$ for non-inertial frames, you can work out the pseudo-force associated with the acceleration of the frame via plugging in the new kinetic energy & potential into the lagrangian and solve for the new equations of motion. $\endgroup$– OblivCommented Feb 20 at 22:41
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$\begingroup$ Thank you for your answer, i have understood why is this so. Thanks for your help! $\endgroup$– bobby76Commented Feb 21 at 16:55