This is from Kleppner and Kolenkow:
What disturbs me is the assertion that every total force over an object of negligible mass is negligible. What I understand for negligible is very small mass compared to forces. I imagine myself hitting a ping-pong ball very strong, and the fact that the ping pong ball has very little mass doesn't imply that the force I exerted is small, but that the aceleration is very large. Of course, if one actually replaces $M=0$ in the equation $\vec{F}=M\vec{a}$, then indeed the force is $0 N$, but i think that, in this case, that the mass equals $0 kg$ is a good deal different than it to be small.
In this example, when we arrive at equation $(2)$, knowing a priori that $F_A>>F_B$, and that $M<<1$, I would state that $a_r$ is very large, rather than claiming $F_A=F_B$. Can anyone explain?
Best Answer
I think what Kleppner & Kolenkow mean is that in real life you cannot instantaneously "apply a force" of a certain finite amount $F$ for a finite time before an object of infinitesimal mass $\delta m$ accelerates away from the influence of the force.
Contact forces are initially zero before contact and gradually increase. Long before the force has increased to the value $F$ which you wish to apply, an infinitesimally small mass will accelerate away from you, preventing the applied force from reaching a finite value. You can never apply more than a negligible force $\delta F$ on a negligible mass $\delta m$ before it accelerates out of your reach.
Even when 2 constant electrostatic "action-at-a-distance" forces held in check, you cannot instantaneously remove one force to leave the other unopposed. Switching off a force takes a finite time, during which the net force $\delta F$ is negligible and causes a finite acceleration away from the source of the force. The smaller the mass $\delta m$ of the object, the smaller the maximum net force $\delta F$ on the object before it escapes the influence of the net force.
In the case of the ping-pong ball, the maximum force which you can apply to it is limited by the deformation of the ball (which occurs in a finite time) and the speed of your arm during contact.