How accurate are physics laws? For example, for newtons' first law $F=ma$, if we can get a measurement of both force, mass and acceleration with a percentage of uncertainly close to $1\times 10^{-9}\%$, will the formula match the value we determined? If not, how many percentages of error could we take and still believe the law still hold?
Newtonian-Mechanics – Understanding the Accuracy of Physics Laws
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Related Solutions
For many (most? all?) physicists, it's something like an axiom (or an article of faith, if you prefer) that the true laws don't change over time. If we find out that one of our laws does change, we start looking for a deeper law that subsumes the original and that can be taken to be universal in time and space.
A good example is Coulomb's Law, or more generally the laws of electromagnetism. In a sense, you could say that Coulomb's Law changed form over time: in the early Universe, when the energy density was high enough that electroweak symmetry was unbroken, Coulomb's Law wasn't true in any meaningful or measurable sense. If you thought that Coulomb's Law today was a fundamental law of nature, then you'd say that that law changed form over time: it didn't use to be true, but now it is. But of course that's not the way we usually think of it. Instead, we say that Coulomb's Law was never a truly correct fundamental law of nature; it was always just a special case of a more general law, valid in certain circumstances.
A more interesting example, along the same lines: Lots of theories of the early Universe involve the idea that the Universe in the past was in a "false vacuum" state, but then our patch of the Universe decayed to the "true vacuum" (or maybe just another false vacuum!). If you were around then, you'd definitely perceive that as a complete change in the laws of physics: the particles that existed, and the ways those particles interacted, were completely different before and after the decay. But we tend not to think of that as a change in the laws of physics, just as a change in the circumstances within which we apply the laws.
The point is just that when you try to ask a question about whether the fundamental laws change over time, you have to be careful to distinguish between actual physics questions and merely semantic questions. Whether the Universe went through one of these false vacuum decays is (to me, anyway) a very interesting physics question. I care much less whether we describe such a decay as a change in the laws of physics.
I think you are confusing systematic and random errors.
Your experimental results can give you no idea about the systematic error.
For example it might be that your timing device is calibrated incorrectly and when the correct time is 1.00 seconds then your timing device gives a reading of 1.10 seconds; when the correct time is 2.00 seconds the timing device gives a reading of 2.20 seconds.
Repeating readings or the smallest subdivision of your scale will not give you an indication of what the systematic error is.
You could only find that error by checking the calibration of your timing device against a reliable standard.
So in this example you have found an estimate of the random error by evaluation the standard deviation and that is the best you can do.
Best Answer
Accuracy can mean different things. While the question asks about the statistical accuracy, what immediately comes when talking about the Newton's laws is that they are non-relativistic, i.e., they are valid up to small corrections of order $v/c$.
Physics laws are based on empirical observations, the symmetries of the universe, and approximations appropriate for a given situation.
Symmetries
For example, we have reasons to think that conservation of momentum or energy are exact laws, since they follow from the symmetry of space in respect to translations in space and time (Noether's theorem). Testing these laws in practice will necessarily result in statistical errors, but improving the precision of measurement is unlikely to uncover any discrepancies.
Approximations
Newton's laws are valid only in non-relativistic limit. Thus, they will hold only up to small corrections of order $v/c$ where $v$ is the speed of the object and $c$ is the speed of light. If our relative statistical precision (in measuring the force, acceleration, etc.) is of order $v/c$, we will observe deviations.
Empirical observations
Laws of thermodynamics are a good example of the laws that were deduced phenomenologycally, as a result of many observations. Yet, statistical physics shows that they hold up to very high precision ($\sim 1/N\sqrt{N_A}$, where $N_A$ is the Avodagro constant). If the precision could be so high or when dealing with systems where the number of particles is not small, we will observe deviations from these laws.
Remark
I recommend the answer by @AdamLatosiĊski, which is technically probably more correct than mine. What I tried to explain in my answer is how the laws of physics are different from, e.g., the biological laws (since the subject was recently debated on this site) - the latter are generalizations of many statistical observations, but not grounded in reasoning about fundamental properties of the universe. They are therefore statistical laws, which are bound to be non-exact. Indeed, even the so-called Central dogma of molecular biology ($DNA\rightarrow RNA \rightarrow Protein$) is broken by some viruses, performing reverse transcription ($RNA\rightarrow DNA$.)