How to know whether a frame is inertial or not?

Question

how one would know that the given frame of reference will be inertial or non inertial, without the use of 'accelerometers'. earlier physicst has defined this concept and I am sure that they define it without the help of accelerometers. I just want to know about the logic behind the whole concept. And I will be not getting any accelerometers in my exam and not able to conduct any experiment in the frame. i want to know how teachers just say that this frame will be inertial or not without any accelerometer.

Answer 1

The problem is, the only way of knowing whether or not you are in an accelerating (non inertial) reference frame is by measurement and/or observation. There’s simply no way around it.

But here is a simple, non quantitative, way to observe if you are in an approximate inertial or non inertial (accelerating) reference frame without an accelerometer. It involves the example of a car. (Note: Though technically not an inertial frame, we will take the Earth frame as being locally inertial because the inertial effects are minuscule for most of our day-to-day experiences and experiments).

Hang an object by a string to the rear view mirror. If the object is hanging perfectly vertically (zero angle with the vertical) you and the car are either at rest or moving approximately at constant speed in a straight line. In other words, you can consider yourself and the car as an inertial reference frame.

On the other hand if the object deflects towards you, away from you, or towards the left or right, or some combination of these, you’re in a non inertial (accelerating) reference frame involving straight line acceleration, braking, and/or turning to the right or left, respectively.

Hope this helps

Answer 2

The simplest way to check to see if a frame is inertial is to check whether momentum and angular momentum are conserved in it.

To check to see whether non-conservation of momentum is equivalent to non-inertiality, note:

$$x_{\rm cm} = \frac{m_{1}x_{1} + m_{2}v_{2}...}{m_{1} + m_{2} + ...}$$

so, therefore:

$$\begin{align} v_{\rm cm} &= \frac{m_{1}v_{1} + m_{2}v_{2}...}{m_{1} + m_{2} + ...}\\ &= p_{\rm net}/m_{\rm net} \end{align} $$

so, if the momentum is conserved, then the velocity of the center of mass doesn't change. But, if this is true in an inertial frame, it cannot be true in any non-inertial frame, because the frame will be accelerating relative to the center of mass of the system. So, therefore, non-inertial frames will always show non-conservation of momentum.