## Thursday, April 26, 2018

### Unizor - Physics - Mechanics - Kinematics - Non Inertial Reference Frames

Notes to a video lecture on http://www.unizor.com

Non-Inertial Frame of Reference

Definition of non-inertial frame of reference is simple. Any frame of reference that is not inertial is non-inertial.

Recalling the definition of inertial frames as those where the Law of Inertia holds true, we can say that non-inertial frame is a frame where the Law of Inertia does not hold true.

To get a feel of non-inertial frames, let's consider a few examples.

Example 1

Let's start with inertial frame of reference XYZ and an object at point P(1,0,0) at rest. There are no forces acting on it, so it stays at rest at the same point P(1,0,0).

Now consider a system of coordinates UVW with an origin coinciding with
an origin of XYZ frame, the W-axis coinciding with Z-axis of XYZ frame
and uniformly rotating (relative to XYZ frame) counterclockwise around
the W-axis with angular speed ω.

It is easily seen that point P in UVW system of
coordinates rotates clockwise around the origin of coordinates within
UV-plane along a circle of radius 1 with an angular speed ω.

So, we have a situation when there are no forces acting upon our object,
yet it rotates in UVW frame along a circular trajectory - definite
disagreement with the Law of Inertia that states that an object,
not acted upon by unbalanced forces, should stay at rest or move with a
constant vector of velocity, which implies that its trajectory must be a
straight line.

Therefore, this UVW frame is not inertial, hence non-inertial.

Example 2

Let's start with inertial frame of reference XYZ and an object at point P(1,0,0) at rest. There are no forces acting on it, so it stays at rest at the same point P(1,0,0).

Now consider a system of coordinates UVW with the origin initially (at t=0)
coinciding with the origin of XYZ frame and all axes initially
coinciding with corresponding axes of XYZ frame. Now assume that UVW
system started to move along the U-axis forward with constant
acceleration a.

It is easily seen that point P in UVW system of coordinates moves along the U-axis in negative direction with constant acceleration a.

So, we have a situation when there are no forces acting upon our object, yet it moves with constant acceleration −a in UVW frame along the U-axis, thus changing its velocity in magnitude - definite disagreement with the Law of Inertia that states that in this case in the inertial frame an object should move with constant velocity (direction and magnitude).

Therefore, this UVW frame is not inertial, hence non-inertial.

Generally speaking, if the origin of UVW frame is not uniformly moving relative to the origin of inertial frame XYZ, while axes of these frames are correspondingly parallel, the frame UVW is non-inertial.

The proof is simple.

If in one frame the position of an object at any moment in time is Pxyz(t) and in another frame the position of the same object is Puvw(t) and the position of the origin of UVW frame in XYZ-coordinates is Qxyz(t) these positions are related as

Pxyz(t) = Puvw(t) + Qxyz(t)

Since XYZ is inertial frame, an object that is not acted upon by
any unbalance force moves along a straight line with constant velocity
vector (that might be a zero vector). That means, the function Pxyz(t) is linear function of time t for each of its component x, y and z.

If the origin of UVW frame is not uniformly moving relative to the origin of inertial frame XYZ, function Qxyz(t) is not linear for its components (otherwise, it would move uniformly).

Therefore, function Puvw(t) cannot be linear, which means an object moves in the UVW in a non-uniform mode.

So, we see that an object, free from all actions by unbalanced forces,
does not move uniformly in UVW frame. Hence, the frame is non-inertial.