Re: The rope

You're confusing momentum with velocity. If the mass is zero (more precisely: negligible), then the momentum is zero even if the velocity is non-zero and changes with time. So the rope can accelerate, and yet the forces on the two ends are equal and opposite. More precisely: their difference is negligible as long as the mass is enough small.

We're simply considering the vector equations (omitting vector notation):

dP(t)/dt = F_A + F_B + G
P(t) = m v(t)
G = - m g e_z

as m → 0, F_A → -F_B.

You, like Newton, are also making additional assumptions that you are not explicitly stating. For example, the rope has thickness, which means that the air pressure generates on it a net non-zero force because of buoyancy. This leads to the two forces at the end not being exactly equal. Yet you didn't state explicitly that you are neglecting air or buoyancy. You're also making the assumption that the gravitational field is constant, which it isn't. Or that it is constant on a horizontal straight plane, which it isn't because the gravitational field has spherical symmetry. Yet you aren't explicitly stating that you're making all these approximations.

Because they're self-understood in the present context.

Likewise, for the readership addressed by the Principia (which wasn't meant as a textbook), the assumption of negligible mass was self-understood.

On 250527 06:54, Luigi Fortunati wrote:

Jonathan Thornburg [remove -color to reply] il 25/05/2025 09:53:48 ha
scritto:

...
The key distinction is that Newton was referring to a situation where
the rope is treated as massless, with no forces acting on it except for
the pulls (tension) at each end.
...

>
It's not a fundamental distinction and Newton never said it.
>
In any case, regardless of what Newton said (or didn't say), the
absence of mass doesn't change anything I wrote
>
The rope has no mass? Okay, it has no mass.
>
Are you saying that there are two tensions at the ends of the rope? Of
course there are!
>
On one side there is the tension caused by the force of the horse and
on the other there is the tension caused by the force of the stone.
>
The horse pulls the rope forward (action) and generates its tension,
the stone pulls backwards (reaction) and generates the other tension.
>
The rope (with or without mass) is in the middle and must move
accordingly: if the horse pulls more the rope *must* accelerate
forwards, if it pulls more the stone *must* accelerate backwards, if
they pull equally it cannot accelerate.
>
So just look at how the rope moves to understand if the action wins or
if the reaction wins or if they are equal.
>
Luigi Fortunati