Inertial Frames of Reference and Newton's First Law
Quick A frame of reference in which Newton's first law is valid is called an inertial frame of reference.
Details The concept frame of reference is central to Newton's laws of motion. Suppose a person is in a train that is accelerating from stop. If the person was in the aisle on roller skates, the person would start moving bacward relative to the train as it accelerates. If the train was coming to a stop, the person would start moving forward down the aisle as the train slowed to a stop. In either case, it looks as though Newton's first law is not obeyed – there is no net force acting on the person yet the person's velocity changes.
The idea is that the train is accelerating with respect to the earth and is not a suitable frame of reference for Newton's first law. This law is valid in some frames of reference, and is not valid in others. A frame of reference in which Newton's first law is valid is called an inertial frame of reference. The earth is at least approximately an inertial frame of reference, but the train is not. It should be noted that the earth is not a completely inertial frame, owing to the acceleration associated with its rotation and its motion around the sun, but these effects are quite small and can be considered negligible. Because Newton's first law is used to define what is meant by an inertial frame of reference, it is sometimes called the law of inertia.
The following figure shows how Newton's first law can be used to understand what is experienced when riding in a vehicle that is accelerating. In the figure, the train car is initially at rest, then it begins to accelerate to the right. A ball has virtually no net force acting on it, since its shape nearly eliminate the effects of friction; hence it tends to remain at rest relative to the inertial frame of the earth, in accordance with Newton's first law. As the train accelerates around it, it moves backwards relative to the train car. In the same way, a ball in a train car that is slowing down tends to continue moving with constant velocity relative to the earth. Thie ball moves forward relative to the car. An automobile is also accelerating if it moves at a constant speed but is turning. In this case a passenger tends to continue moving relative to the earth at constant speed in a straight line; relative to the vehicle, the passenger moves to the side of the vehicle on the outside of the turn.
In each case shown in the figure, an observer in the vehicle's frame of reference might be tempted to conclude that there is a net force acting on the passenger in each case, since the passenger's velocity relative to the vehicle changes in each case. This conclusion is simply wrong; the net force on the passenger is indeed zero. The vehicle observer's mistake is in trying to apply Newton's first law in the vehicle's frame of reference, which is not an inertial frame and in which Newton's first law isn't valid.
The earth's surface is only one (approximately) inertial frame of reference - there are many inertial frames. If there is an inertial frame of reference A, in which Newton's first law is obeyed, then any second frame of reference B will also be inertial if it moves relative to A with constant velocity
The velocity of particle B is represented by the variable
v
B. But, the velocity of particle B is with respect to the origin O, so the variable can be rewritten as
v
B/O. The same can be said for the velocity of particle A; it is with respect to the origin, so that the variable
v
A can be rewritten as
v
A/O. Eq1 then becomes:
(Eq2)
v
B/O =
v
A /O +
v
B/A
Consider the following figure in reference to the preceding and following writings:
Suppose that the body B moves with constant velocity
v
B/O with respect to an inertial reference frame, origin O. By Newton's first law the net force on this body is zero. The velocity of B relative to another frame A has a different value:
v
A /O =
v
B/O −
v
B/A
But if the relative velocity
v
B/A of the two frames is constant, then
v
A /O will be constant as well. Thus the origin A is also an inertial frame; the velocity of B in this frame is constant, and the net force on B is zero, so Newton's first law is obeyed in A. Obervers in frames O and A will disagree about the velocity of B, but they will agree that B has a constant velocity (zero acceleration) and has zero net force acting on it.
There is no single inertial frame of reference that is preferred over al others for formulating Newton's laws. If one frame is inertial, then every other frame moving relative to it with constant velocity is also inertial. Viewed in this light, the state of rest and the state of motion with constant velocity are not very different; both occur when the vectorsum of forces acting on the body is zero.
