Unizor - Physics - Mechanics - Inertial Frame of Reference

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



Inertial Frame of Reference



The term frame of reference in the context of Mechanics is understood in the same sense as the system of coordinates.

Since coordinates depend on the system of coordinates, the position functions we were dealing with (x(t), y(t) and z(t)) are obviously related to a system of coordinates established in our three-dimensional space.



Assume, for example, that in one system of XYZ-coordinates (or frame of reference, or just frame) our position functions are

x(t) = t²

y(t) = 0

z(t) = 0

Consider a new frame with UVW-coordinates originated at point (3,0,−5)
in the old frame with axes correspondingly parallel to those of the old
frame.

In this new frame the coordinates will be different:

u(t) = t²−3

v(t) = 0

w(t) = 5



So, when we talk about position functions or their derivatives (velocities, accelerations), we always have in mind some frame of reference or system of coordinates, relative to which the position is defined.



In philosophical sense the position of any object is absolute, but, if
we want to express this position in some numerical way, it's always
relative to some chosen frame of reference.

If the frame of reference is not defined (at least, implicitly, like,
when we say that the speed of a car is 60 km/hour, we mean "relative to
the road", which defines a frame of reference), we cannot quantitatively
talk about position and other quantitative characteristics like
velocity or acceleration.



Obviously, we would like to have an absolute frame of reference, which is absolutely
not moving anywhere and can be used as The Main Frame of Reference,
relative to which all other objects and frames can be referenced.

Alas, there is no such absolute reference frame because everything in this Universe is moving somewhere somehow relative to something.

However, the next best thing would be a reference frame tied to position
of stars, because they are so far away that seem standing still. This
reference frame we will keep in mind in those rare cases when we mention
motion without explicitly or implicitly referring to any particular
frame.



Imagine a comet in space flying far from any solar system, so there are
almost no gravitation fields around, no any other forces that can change
this comet's course. Relative to stars that are almost standing still,
this comet moves almost in the same direction with almost constant
velocity. This is an almost uniform motion relatively to almost steady
reference frame and the best approximation to an abstract concept of uniform motion in an inertial reference frame.

We have just introduced a new term inertial reference frame that we explain below.



The almost uniform motion of a comet far from any forces of gravity is the basis for the Law of Inertia that should be considered as an axiom confirmed by experiments. It states that an object at rest stays at rest and an object in uniform motion stays in this uniform motion (that is, no change in velocity vector), unless acted upon by unbalanced forces.



Inertia is the property of an object to stay at rest or maintain its uniform motion in the absence of unbalanced forces.
That's why the Law above is called the Law of Inertia, and that's why
the frame of reference related to almost immovable stars where this
phenomenon takes place is called inertial frame of reference.



The most difficult part in understanding of this Law of Inertia is to understand the meaning of the state of rest or uniform motion
since no system of coordinates is mentioned. Indeed, if in one
reference frame an object moves uniformly with some velocity vector, in
another system, moving relatively to the first, the velocity vector,
represented as derivatives of its coordinates, looks differently and not
necessarily be a constant, as required in uniform motion.



We can overcome this difficulty and say that the one and only reference frame implied in the Law of Inertia
is the one based on almost immovable star-based frame. That's good
enough as the first step, but we will always want to use different
systems in solving different mechanical problems and in most cases it's
inconvenient to use this system. For example, if a car moves along a
straight road, we would like a reference frame originated at the car's
starting point on Earth and X-axis going along the road, assuming it's a
straight line and not too long, so we can ignore the curvature of our
planet.



But here is an interesting consideration. Assume, an object is in
uniform motion relative to some XYZ frame of reference (say, our
star-based one) and another UVW frame of reference is in uniform motion
relatively to the first one with corresponding axes parallel to each
other.

The uniform motion of an object in the XYZ frame, as we know, can be expressed as

x(t) = a·t + x₀

y(t) = b·t + y₀

z(t) = c·t + z₀

Or in vector form

P(t) = t·V + P₀

where the velocity vector

V = {a, b, c}

and initial position

P₀ = {x₀, y₀, z₀}



When the UVW frame uniformly moves relative to XYZ frame with correspondingly parallel axes according to some velocity vector Ω , coordinates of the same object in different frames of reference are related as linear functions of time.

If in one frame the position of object at any moment in time is P_xyz(t) and in another frame the position of the same object is P_uvw(t), these positions are related as

P_xyz(t) = P_uvw(t) + t·Ω + Q₀

where Ω is a velocity vector of the origin of the UVW frame relative to XYZ frame and Q₀ is the initial position of the origin of the UVW frame relative to XYZ frame at time t=0.



Here is the logic behind this equation.

At time t=0 an object is at position P₀ = P_xyz(0) in XYZ frame.

At the same time it is at position P_uvw(0) in UVW frame, which is shifted by vector Q₀ in XYZ frame from its origin.

Since a vector P_xyz(0) from the origin of XYZ frame to an object's position at time t=0 equals to a sum of a vector from the origin of XYZ frame to the origin of UVW frame Q₀ and a vector from the origin of UVW frame to an object P_uvw(0), our equation is true for the initial time moment t=0.



As the time goes, the origin of the UVW frame moves to Q₀ + t·Ω . If the position of an object in this UVW frame is P_uvw(t), we can express the position in XYZ frame P_xyz(t) as a sum of a vector to a new position of UVW's origin and P_uvw(t), which gives the formula suggested above:

P_xyz(t) = P_uvw(t) + t·Ω + Q₀



Differentiating both sides, we will get an equation about velocities of our object in two frames of reference:

V_xyz(t) = V_uvw(t) + Ω

This equation shows that if the vector of object's velocity is constant in one frame, it's constant in another since vector Ω , characterizing the uniform movement of frame UVW relative to frame XYZ, is constant.



Consider now any frame of reference moving uniformly relative to a star-based frame. Since the Law of Inertia
is an axiom in the star-based frame, we can now say that the object,
moving uniformly in this system, moves uniformly in any other system
that uniformly moves relative to a star-based.

That's why any frame of reference moving uniformly relative to a star-based frame, is also an inertial frame, since the Law of Inertia is true in it.



So, we have postulated that the Law of Inertia is true for star-based frame. Based on that, we called this coordinate system an inertial frame of reference. Then we have proved that in any other frame of reference, moving uniformly relative to a star-based one, the Law of Inertia is true as well, which allowed to call that other frame also an inertial frame of reference.

As well as our chosen system of coordinates moves approximately uniformly relative to a star-based one, it can be called an inertial frame of reference and the Law of Inertia is expected to be true in it (within certain precision, of course).