# Inertial frames?

This is from a book.

A traing is moving on earth. A ball is sitting at rest (relative to the train) on the floor of the train which is moving at constant velocity relative to a tree. If the contact between the ball and the train floor is frictionless, the ball receives no net external force. Observed from the tree-frame, the ball will continue to move at the same velocity as the train. Observed from the train-frame, the ball will continue to stay at rest. So the Law of Inertia holds in both the tree- and train-frames. They are both inertial frames.

But the earth itself rotates and it also orbits the sun. So, the earth is not inertial frame. So, why do we lable the train and the tree 'inertial frames'.

• The ugly truth of science that they don't tell you about: "approximation". :P – Dvij Mankad Apr 3 '19 at 14:40
• @DvijMankad That's only approximately true – Aaron Stevens Apr 3 '19 at 14:43
• @AaronStevens :) – Bill N Apr 3 '19 at 15:04

Earth is non-inertial frame but its considered as inertial when the regions are too small to consider the gravitational effects due to sun and acceleration of earth around sun. So on smaller scale(as compared to other objects in space) we can treat earth as inertial frame so consequently the other frames which fulfil the condition of being inertial with respect to this frame are considered inertial too.

It is obvious that the text is choosing to neglect the rotation of the Earth. The surface of the Earth is approximately an inertial frame, so we usually treat it as such when we don't want to get into the technicalities you discuss.

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As pointed out by @DvijMankad: The orbital motion around the Sun is not in need to be approximated because the motion of the Earth around the Sun is indeed a free-fall motion and thus, it doesn't do any harm to the inertial character of a frame attached to the Earth. In fact, it ensures it. If the Earth were held fixed in the sense of getting obstructed from following a free-fall motion, that would create an issue in treating the Earth as an inertial frame.

• The rotation of the Earth would need to be neglected to assert what has been quoted. But, the orbital motion around the Sun is not in need to be approximated because the motion of the Earth around the Sun is indeed a free-fall motion and thus, it doesn't do any harm to the inertial character of a frame attached to the Earth. In fact, it ensures it. If the Earth were held fixed in the sense of getting obstructed from following a free-fall motion, that would create an issue in treating the Earth as an inertial frame. Correct me if I am mistaken here. – Dvij Mankad Apr 3 '19 at 14:46
• @DvijMankad You are talking from a GR perspective? – Aaron Stevens Apr 3 '19 at 14:49
• Yes, of course, I mean otherwise there is no place to argue about what is inertial and what is not. In Newtonian world, we can only state that we did an experiment and found out that a frame is inertial xD And then it is riddled with all these paradoxes as to why the ISS orbiting the Earth can also behave as an inertial frame when the Earth itself is an inertial frame and so on. I know one can take the Earth as a true inertial frame and explain things in the ISS with pseudo forces but one can do the same thing other way around and there is no winning either way. – Dvij Mankad Apr 3 '19 at 14:52
• @DvijMankad This is true. I have put the relevant parts of your first comment into my answer. Thanks – Aaron Stevens Apr 3 '19 at 14:55
• @DvijMankad In Newtonian world, we can only state that we did an experiment and found out that a frame is inertial I don't agree. Newtonian edifice is theoretically consistent and you can explain why it's legitimate to treat Earth as an inertial frame (as far as its orbital motion is concerned - rotation aside). It's a logical consequence of just one experimental fact: that gravitational force is proportional to mass. – Elio Fabri Apr 3 '19 at 19:54

Including a spinning earth, if the train is travelling purely in the east-west direction the ball will stay in place. However, if the train has a velocity component in the north-south direction, this is not generally the case. The coriolis force will then act to move the ball with respect to the train. This is basically due to the fact that the surface of the earth move at different velocity as compared to the axis of rotation. When you go north on the northern hemisphere you will be getting closer to the axis of rotation acting as to push the ball on the train eastwards. As the book assumes that the contact between the floor and the ball is "frictionless", you can not really ignore the coriolis force.

• if the train is travelling purely in the east-west direction the ball will stay in place Only if the train is travelling along equator. As to friction, it doesn't forbid a ball to roll. – Elio Fabri Apr 3 '19 at 15:37
• If you assume that the inclination of the floor initially is such as to counterbalance both the gravitational force and the centrifugal force, then the ball will stay put if the train is moving west or east, even if not at the equator. – Agerhell Apr 3 '19 at 16:03
• ...counterbalance both the gravitational force and the centrifugal force ... Coriolis force depends on train's velocity wrt ground. So it can't be counterbalanced that way. In an inclined railway like that the train would still be subject to a tangential force, northward or southward, varying with latitude as $\sin\phi$. – Elio Fabri Apr 3 '19 at 19:45