# Why every force on negligible mass is negligible?

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?

## 2 Answers

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.

• I think I get the point. Let me apply your argument to the astronauts' example and correct me if I'm wrong: If Alice exerts a considerable bigger(i.e., non-negligible) force than Bob, then, given the little mass of the rope, the rope should accelarate a lot in Alice's direction. Given that Alice has a lot more mass than the rope, the reaction force on her doesn't accelerate her too much, and in fact, it pushes the opposite direction than the rope is going. Same with Bob, whose aceleration would be in the direction of the rope, but much smaller. Commented Sep 19, 2016 at 6:35
• Then we can't keep the condition that both keep holding the rope, since the rope moves faster than each, and is, of course, of finite lenght. Commented Sep 19, 2016 at 6:35

I think that instead of

What I understand for negligible is very small mass compared to forces.

You meant

What I understand for negligible is very small mass compared to other masses.

You are correct that the statement

the total force on any body of negligible mass must be vanishingly small

is not a general principle which is always true and your example of the ping pong ball is a good one.
Another example is the free fall of the ping pong ball.
In both these examples one would say that the acceleration of the object producing the force (ping pong bat / Earth) is negligible because its mass is so much greater than the mass of the object on which they are applying the force.

The idea of an ideal rope/string which is massless and inextensible is used to simplify problems and the statements made in the text that you have provided relate to situation where you are interested in the motion of a large mass.
The ping pong example is one where you are interested in the motion of the object with a small mass.