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But I have some comment on the basic statement in the text itself:
"...in order for the object to maintain its shape, different parts
will need to accelerate at different rates".
This statement is ambiguous. Its truth value depends on the chosen
reference frame (RF) and on definition of "shape".
Let us define the
shape as an instantaneous configuration of the object in a given RF.
Then it is easy to see that in the initial rest frame of the object
(frame A), the statement is wrong.
Different acceleration rates for
different parts will surely distort the shape.
In order to conserve
it in A, we need the same acceleration a for all parts.