*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

**at rest. There are no forces acting on it, so it stays at rest at the same point**

*P(1,0,0)***.**

*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

**in UVW system of**

*P*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

**at rest. There are no forces acting on it, so it stays at rest at the same point**

*P(1,0,0)***.**

*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

**in UVW system of coordinates moves along the U-axis in negative direction with constant acceleration**

*P***.**

*a*So, we have a situation when there are no forces acting upon our object, yet it moves with constant acceleration

**in UVW frame along the U-axis, thus changing its velocity in magnitude - definite disagreement with the**

*−a***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

**and in another frame the position of the same object is**

*P*_{xyz}(t)**and the position of the origin of UVW frame in XYZ-coordinates is**

*P*_{uvw}(t)**these positions are related as**

*Q*_{xyz}(t)

*P*_{xyz}(t) = P_{uvw}(t) + Q_{xyz}(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

**is linear function of time**

*P*_{xyz}(t)**for each of its component**

*t***,**

*x***and**

*y***.**

*z*If the origin of UVW frame is not uniformly moving relative to the origin of

*inertial frame*XYZ, function

**is not linear for its components (otherwise, it would move uniformly).**

*Q*_{xyz}(t)Therefore, function

**cannot be linear, which means an object moves in the UVW in a non-uniform mode.**

*P*_{uvw}(t)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*.

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