However, this is not always the most convenient frame: for someone sitting in a moving airliner, in an orbiting spaceship or upon the rotating Earth, it is much easier to observe the way an object moves relative to its immediate surroundings than to figure out its motion relative to the distant universe! It is therefore often much preferable to derive the corrections which must be applied to the laws of physics in the local frame of reference, and then by taking those corrections into account, calculate the local motion. |
Two rather typical cases are examined, here and in the next section. In those cases, relative to the rest of the universe, the local frame--
(1) Moves with constant velocity in a straight line
(2) Rotates at a constant rate around a fixed point.
Constant velocity in a straight line
Suppose we sit in an airliner (or in a train, or on a ship) that moves with a velocity v0 , constant in direction and magnitude ("uniform motion").
Strictly speaking, this constancy should be with respect to the "absolute frame" of the distant universe. Here, however, we shall only assume constancy relative to the surface of the Earth, and later show that this gives a pretty good approximation to the "absolute frame."
How do observations differ in the two frames--the airliner cabin and the Earth? Very simply: any velocity v measured inside the airliner corresponds to a velocity
v' = v + v0
with respect to the ground (in the most general case v', v and v0 are vectors and their directions may also differ).
And how about accelerations? Compare:
--the acceleration a with respect to the cabin
However, because v and v' = v + v0 differ only by a constant velocity v0, the two change at exactly the same rate, so that a = a'. For the falling penny, both v' and v grow at the same rate g (about 9.8 meter/sec each second); the fact that a big constant velocity of 600 miles/hr is included in v' but not in v does not add anything to a', because "constant " means unchanging.
is the rate at which v changes.
--the acceleration a' with respect to the ground
is the rate at which v' changes.
And because the accelerations are the same, so are the forces:
In the frame of the plane F = ma
and since a = a', it follows that F = F'. In short:
In the frame of the Earth F' = ma'
In other words: Newton's laws can be used in uniformly moving frames, as if those frames did not move. The theory of relativity modifies those laws when the magnitude of v or v0 approaches the speed of light in vacuum, but all motions studied here are much slower.
All laws of mechanics remain the same in
a frame moving at a constant velocity.
Now about the frame of the Earth: it is not the same as the frame of the distant universe. The rotation of the Earth around its axis has a small effect, most of which can be taken into account by a slight modification of the value of g (see next section). The accelerations added by the Earth's motion around the Sun (which were neglected) are counteracted by the Sun's gravity pull (also neglected), and both are small. It is thus not a bad approximation to regard the Earth as the ultimate frame of reference.