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In the previous article, we looked at coordinate systems and how to specify the position of a particle relative to the origin of a coordinate system.  Now, the choice of coordinate system is not unique.  Indeed, one could envisage a system such as that shown in the figure below.  In this case, we have two coordinate systems, with origins labelled O and O^’, and it is possible to specify the position of the particle with respect to either coordinate system.  The positions are of course different in the two cases, the difference being the offset of the origins of the coordinate systems.

The position of the particle in coordinate system O is related to the position in O^’ through the relation

where of course R is the vector displacement of O and O’.

From now on, we will think of two observers, one located in coordinate system O and the other located in O^’.  The observer in O sees the position of the particle as r and the observer in O^’ sees the position as r^’.  The velocity of the particle, as seen by the observer in O, is given by

where i, j and k are unit vectors defined for coordinate system O.  Similarly, the velocity of the particle as seen by the observer in O^’ is

where in this expression the primed quantities refer to coordinates and unit vectors defined in O^’.

The particle velocities for our two observers are related by

provided that the x axes in O and O^’ are parallel, and similarly for the y and z axes.  If the frame of reference O^’ does not rotate, then this can always be chosen to be true.  The case of a rotating frame of reference will be considered in a later article.  Under the conditions of no rotation, the particle accelerations for our two observers are related by

with similar definitions for the other quantities.

Now, provided that the frame of reference O^’ is not accelerating relative to O, A = 0 and

The acceleration of the particle is the same for both observers, and this leads to the notion of an inertial frame of reference.  That the accelerations are seen to be the same in both frames further leads to the notion that inertial frames are equivalent, in the sense that the laws of physics (in this case the particle acceleration and hence Newton’s laws of motion) are the same in all inertial frames of reference.

Now suppose that A is not zero.  If the particle has mass m, then Newton's second law for the observer in O takes the form

Now, using the expression above for acceleration, this can be rewritten in the form

This is a very revealing expression, even though it is algebraically trivial to derive.  It states that the observers in O and O^’ have differing views about the nature of the forces acting on the particle.  In particular, if the observer in O sees a force F acting on the particle, the observer in O^’ sees a force that is different by an amount mA.

The “force” mA is often called an “inertial” or “fictitious” force, because it does not arise due to some external agency acting on the particle.  Rather, it arises purely as a result of the nature of the coordinate system from which the motion is observed.

An example will illustrate this further.  (This example is taken from Analytical Mechanics by Fowles, 1986).  Consider a block of wood of mass m lying on a larger block of wood, and assume that the larger block can be made to accelerate at some rate A along a table.  Let the coefficient of sliding friction of the block of wood on the table be µ.  At what acceleration A does the block of wood start to slip on the table?

This situation is illustrated in the following diagram:

In the terminology used previously, the coordinate system O is fixed on the table and the system O^’ is located on the larger block of wood.  The only force acting on the smaller block in the horizontal direction is the friction force f.  In coordinate system O, the equation of motion is therefore

so that as far as the observer in O is concerned, the block of wood experiences an acceleration to the right of magnitude f/m.

For the observer in O^’, the equation of motion is

The smaller block will start to slip when the value of mA exceeds the maximum value of f, which is µmg.  At this point, the acceleration  begins to take on a negative value, and to the observer in O^’, the block starts to accelerate to the left.  The critical value of A is therefore

This is the acceleration at which the smaller block starts to slip.  Note that at this point, the acceleration (to the right) of the block as seen by the observer in O is also µg, as would be expected.

Next:  Rotation and Angular Velocity