Physics Asked by miguel.negrao on January 18, 2021
It is said the tractrix is the curve described by a mass being pulled by a string, where the end of the string being pulled moves with constant speed, and the mass suffers a friction force. What is the physics explanation for why in the tractrix the velocity is always aligned with the string pulling the mass. Why if $h(t)=(h_x(t),h_y(t),h_z(t))$ is the position of the mass, and $j(t)$ is the position of the start of the string pulling the mass, then the velocity is always aligned with the string, that is $h'(t) = k (j(t) – h(t) )$ holds, where $k>0$ is some constant. Can this be derived by for instance stating the forces applied on the mass and then using $F = m a$ or some other physical argument ?
You can imagine it as you pulling your dog (object) with a rope (string), while walking on a straight line, but the dog don't want to follow where you are pulling him and just want to stay in place. As you move, the tension of the rope felt by the dog is always directed toward you, making the dog move slightly (but unwillingly) toward the direction of that tension.
Answered by philip_0008 on January 18, 2021
UPDATE:
Many thanks for the update, Miguel. I am sorry, I misunderstood the description of the curve, which is misleading. The "mass" has no inertia (now I understand what philip_0008 meant in his comment) and is pulled infinitessimally slowly. So the "pulling" here is a "quasi-static" process, not a dynamic one. Although there is a force, it is at all times balanced by the drag, so there is no acceleration, and F=ma cannot be applied. There is also no velocity in the usual physical meaning of the term.
ORIGINAL ANSWER:
The velocity of the pulled object (or of the pulled end of the string) does not necessarily have to be in the same direction as the string. For example, the particle could be in circular motion. So there is no point trying to prove mathematically that velocity is in the direction of the string. I suggest that you provide a reference for this claim.
Perhaps what you have read is that if the only force on the particle is the tension in the string (eg the particle moves on a frictionless horizontal plane), then the vector acceleration is always in the direction of the string, because that is the direction of the net force. But if drag is included then this is not generally true either.
Answered by sammy gerbil on January 18, 2021
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