# Newtonian Mechanics – Should Pseudo-Forces Be Applied Constantly Due to Earthâ€™s Non-Inertial Frame of Reference?

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Clearly, the Earth is accelerating (as it has centripetal acceleration), and thus it is a non-inertial frame of reference. Then, why don't we use pseudo forces all the time? My teachers told me that the effect produced by the centripetal acceleration of the Earth on our calculations is negligible and hence we can safely neglect pseudo force. But how? I want rigorous proof with examples that show exactly how much we are compensating in exchange for neglecting that the Earth is a non-inertial frame of reference.

The way to properly answer your question is, for lack of a better word, by enumeration. Consider all the ways the Earth moves, compute the fictitious forces corresponding to them and check that they are negligible for the problem at hand. "All" is a strong word, but we can do quite well by looking at what goes into the expressions so that we can tell which ones are the most relevant.

The various motions can be all interpreted as compositions of rotations, so we can use the expression listed at the end of this Wikipedia section: the three kinds of fictitious force acting on a body of mass $$m$$, moving at velocity $$v$$, at a distance $$r$$ from the origin of the rotation (which happens at angular velocity $$\Omega (t)$$) are

• the Coriolis force, whose maximum magnitude is $$2m \Omega v$$,
• the centrifugal force, whose maximum magnitude is $$m \Omega^2 r$$,
• the Euler force, whose maximum magnitude is $$m \dot{\Omega} r$$ (the dot denotes a time derivative).

I'm saying "maximum" so we have an upper bound, which is attained when the vectors appearing in the vector products are perpendicular. Now, what is left is to take the system at hand, estimate the forces, and finally see how they compare to the precision of your experiment.

As an example, consider the rotation of the Earth around its axis. An upper bound for $$r$$ is the radius of the Earth, $$r \approx 6400 \mathrm{km}$$. The angular velocity is typically $$\Omega \approx 2 \pi / (1 \mathrm{d}) \approx 7\times 10^{-5} \mathrm{Hz}$$. The typical value of the variation of $$\Omega$$ is harder to estimate - it fluctuates a lot, from a quick look at the data here it seems like the variation of the length of a day to the next can be of the order of $$100\mathrm{ms}$$ at most. This relates to a variation of angular velocity of the order of $$2 \pi (100 \mathrm{ms}) / (1 \mathrm{d})^2 \approx 10^{-10} \mathrm{Hz}$$, therefore the quantity $$\dot{\Omega}$$ will be $$\approx 10^{-10} \mathrm{Hz / d}$$.

The velocity is dependent on your specific experiment: for concreteness' sake, let us assume we are considering things moving at $$10 \mathrm{m/s}$$ at most (as much as a fast cyclist, for instance). With these numbers, let us compute accelerations instead of forces (removing the $$m$$, so that our considerations generalize for any mass):

• the Coriolis acceleration will be at most $$\approx 10^{-3} \mathrm{m/s^2}$$,
• the centrifugal acceleration will be at most $$\approx 3 \times 10^{-2} \mathrm{m/s^2}$$,
• the Euler acceleration will be at most $$\approx 6 \times 10^{-9} \mathrm{m/s^2}$$.

This procedure can be applied for any other kind of rotation you are worried about --- for example, the rotation of the Earth around the Sun will have $$\Omega \approx 2 \pi / (1 \mathrm{yr})$$ and $$r \approx 1 \mathrm{AU}$$; this leads to a Coriolis force of the order of $$\approx 4 \times 10^{-6} \mathrm{m/s^2}$$ and a centrifugal force of the order of $$\approx 5 \times 10^{-3} \mathrm{m/s^2}$$.

From this analysis we can already tell that the most significant force to consider is the centrifugal one from the Earth's rotation, followed by the centrifugal one from the Earth's revolution, et cetera. If you had large velocities (for example, if you were doing ballistics) then the Coriolis contribution might be the largest.

So, take the experiment you want to perform, make some Fermi estimates like these, and see whether you can experimentally detect these effects! If it turns out you can, you should look at how exactly they affect your object since they are quite different: the full aforementioned expression shows how to determine their direction as well, and sometimes you might not be able to disentangle their contribution from something else. For example, if you are on the Equator the centrifugal force points radially outward, so it is equivalent to a decrease in the effective gravity at that point.

On the other hand, the Coriolis force deflects the motion of a particle, so it is easier to detect - it is the driving principle behind the Foucault pendulum. This experiment also illustrates the point that, while the accelerations are generally small, if you let them act for a long time their cumulative effect can be rather large.

This procedure (estimating the typical size of the corrections) is quite general: it can also be applied to check whether it makes sense to use Newtonian mechanics in your case, or if you should account for general relativity, quantum mechanics or some other more complicated theory.