Suppose I have an Atwood machine, that is, two different masses connected with an inextensible, massless rope over a pulley. Assuming no friction between the rope and the pulley, the heavier mass will accelerate towards the ground, the lighter mass will accelerate towards the pulley, and the rope will accelerate towards the heavier mass. These three accelerations will be equal in magnitude. But this makes no sense to me. Force causes acceleration. But there is no force acting on the rope. And even if there was, the acceleration of the rope would be infinite because its mass is 0. So why does the rope accelerate? And how can the magnitude of this acceleration be finite?
When the (inertial) mass is zero, then the acceleration can be non-zero for zero force.
This is similar, conceptually, to what has been discussed recently regarding an ideal conductor.
Consider Ohm's Law:
$$V = IR$$
Now, what if $R = 0$ as is the case with an ideal conductor?
Clearly, the voltage must be zero for any current. The current through the conductor, then, is determined by constraints external to the ideal wire, i.e., by whatever the ideal wire is connected to.
Consider Newton's 2nd Law:
$$F = ma$$
Now, what if $m= 0$ as is the case with the massless rope?
Clearly, the force must be zero for any acceleration. The acceleration, then, is determined by constraints external to the massless rope, e.g., the attached masses.
Yes, the massless rope is ideal and, thus, not physical but, there can be effectively massless ropes just as there can be effectively ideal conductors. Which is to say that, to the precision one is working to, the rope has zero mass and zero force acting on it but non-zero acceleration.
I agree with Brandon Enright's comment. But even if there were massless ropes, if m=0 and F=0, then F=ma still would hold for any finite a.