# Movement of an object tied to another

Suppose we have two objects $$A$$ and $$B$$, where $$B$$ is tied by a rope to $$A$$ with the slack pulled out, the initial tangential speed $$c'(0)=(0,1)$$ is perpendicular to $$c(0)$$ and $$c(0) = (1,0)$$, and assume that $$A$$ moves minimally as it "pulls" $$B$$, so for example it has a far larger mass. Suppose also that we can ignore any other forces acting on these two objects besides the force of the rope pulling on $$B$$. I think we can then parametrize the movement of $$B$$ by a curve at time $$t$$ as

a $$c(t) = r(t) (\cos(\varphi(t)), \sin(\varphi(t)))$$,

and the force of the acting on the rope is always in the direction of the center, so we can describe $$c''$$ as

b $$c''(t) = k(t) (\cos(\varphi(t)), \sin(\varphi(t)))$$.

If we just "let" things be, $$B$$ will orbit around $$A$$ at a constant speed in a circle. Suppose however that we pull the rope with some additional force so that $$B$$ starts getting closer to $$A$$, or that we start pulling it at a fixed speed (I'm not quite sure what additional force would correspond to this, but I suppose it might need a strong/infinite impulse at the start, so that the change of speed is immediate? I'm not even really sure this makes sense/is possible, hence me asking this question). I imagine all of this as a friction-less pulley system at $$B$$, with us pulling the rope perpendicularly to the orbiting plane.

Two observations I've made, which I would appreciate if someone checked the validity of:

If $$c'\cdot c'' = 0$$ were to be true and also $$c' \neq 0$$, then $$c \cdot c' =0$$ by b, and so the orbit must be a sphere. This means that if $$r'$$ is not zero, then $$c' \cdot c'' \neq 0$$, and so the magnitude of $$c'$$ will be changing.

Expanding out $$c''$$, if b is to be true, then we can acquire the condition: $$2 r'(t) \varphi'(t) + r(t) \varphi''(t) = 0$$. If $$\varphi$$ were to be linear, this reduces to $$C r'(t) = 0$$, again showing that the orbit is a sphere - this means that the angular speed will also be changing.

I think my methods of showing both of the above might be unnecessarily complicated and I would appreciate a simpler way to show it. Especially the second seems intuitively true - based on the fact that the speed will be increasing and the radius getting smaller, but I don't know how to show it.

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Lastly, how exactly can $$r(t)$$ look - assuming we can pull with any amount of force at any time, but only in the direction towards $$A$$? What if the rope can only ever get shorter? Is it possible for $$r'$$ to be constant? Is it possible to satisfy $$r'(0) = 0$$, and then $$r'(x) = -1$$ on $$\langle \epsilon, 1 \rangle$$ for some $$\epsilon > 0$$ ? Assuming $$r'$$ is constant, and using b and $$2 r'(t) \varphi'(t) + r(t) \varphi''(t) = 0$$, then for say $$r(t) = 1-t$$, we get $$\varphi' = C(1-t)^{-2}$$, and so overall $$\varphi = C(1-t)^{-2} + D$$. Maybe we could now show that from this it follows that the force is always "pulling" $$B$$ towards $$A$$, at least while $$t \in \langle 0,1\rangle$$ that is. That should be equivalent to showing that $$k(t)$$ is always negative.

Also, I would like to know whether these sorts of problems couldn't be solved without using forces etc, and simply using only the knowledge of the starting tangential velocity of $$B$$, and the constant velocity of $$r'$$. Maybe we could think of these things in terms of more along "the object is moving at a tangential velocity, the rope is pulling at a constant velocity, and so we simply sum the velocities in some way" - maybe we could approximate the movement of $$B$$ at certain intervals $$0+n \epsilon$$. Working with the position of $$B$$ as a sequence $$p(n)$$ we would approximate the "previous velocity" by $$p(n+1) - p(n)$$, project this "velocity" onto the plane that is normal to the rope, and then finally add the velocity by which the rope is shortening. Something like that.