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The law of inertia states that "if the net force acting on a body is
zero then the velocity remains constant". I do not see why this should
not be applicable in the rotating frame of reference. If the bead on
[or inside of] the rotaing straw does not accelerate then the net
force acting on it is zero.
... The bead won't stay put on the rotating straw unless you provide
an unbalanced force (glue). ...
A frame of reference is inertial if it does not accelerate with
respect to a frame which is already known to be inertial [distant
stars as used by Newton]. What is wrong with this operational
definition?
Nothing! This is a frame in which F=ma seems to work; but this is
not self evident a priori.
It requires that an experiment be done which refers to the world
outside the frame. You ought to be able to determine whether a
frame is inertial or not without referring to the outside world.
Do you think the fixed stars affect the experiment you do in
your laboratoty? To the extent that they do not, the decision
about whether a frame is inertial or not should be within the
capability of the experimenter to determine wholly within the
laboratory.