The First Law & Inertial Frames

A football left on the grass stays put. Kick it on ice and it keeps gliding. Newton’s first law turns that everyday observation into a precise statement, and in doing so tells us which observers we can trust.

Why the laws are subtle#

Newton’s laws connect the forces on a body with the way it moves. To state them perfectly we would first need exact definitions of mass, force and acceleration, yet those definitions themselves lean on the laws. So the laws are partly definitions. Rather than untangle that history, we take a practical route: assume we know how to assign a mass to a body, a size and direction to a force, and how to measure acceleration in a chosen frame (as in the previous chapters), and then learn how to use the laws.

The first law#

The “if and only if” works both ways. Know that the forces cancel, and you know the particle is not accelerating. Know that it is not accelerating, and you know the forces cancel.

Which observer is right?#

Rest and acceleration only mean something relative to a frame, and different frames measure different accelerations. Can the first law hold in all of them? Try it.

Suppose the lift’s cable snaps. The cabin and everything in it fall with acceleration 9.8 m/s². Two careful observers apply the first law to the hanging lamp:

• Inside the lift, the lamp is at rest. Conclusion: the forces on it cancel, so $T = W$.
• On the ground, the lamp accelerates at 9.8 m/s². Conclusion: the forces do not cancel, so $T \ne W$.

Both cannot be true. One of the two frames is a “bad” frame in which the first law simply fails. (Here the cord goes slack, so $T = 0$ and the ground observer is right.)

Read this way, the first law becomes a definition of an inertial frame. Why call it a law, then? Because nature provides one: for everyday purposes the Earth is an inertial frame. A book resting on your desk has forces that (to excellent accuracy) cancel. A very precise measurement would reveal a tiny imbalance, since the Earth spins and orbits, but for routine problems the Earth frame works. Newton noticed exactly this and stated it as his first law. Unless we say otherwise, we always work in an inertial frame.

Other inertial frames#

Let S be inertial and let S′ move relative to S with constant velocity. From the previous chapter, accelerations measured in the two frames are related by

and $\vec a_{S',S} = 0$ because S′ moves uniformly. So $\vec a_{P,S'} = \vec a_{P,S}$: both frames measure the same acceleration, and since the first law holds in S, it holds in S′ as well.