A scenario of non-uniform circular motion

[Dark85]

TL;DR

I was wondering about whether non uniform circular motion can exist?

(All the needed diagrams are posted below)
My friend came up with the following scenario.

Imagine a fixed point and a perfectly rigid rod of a certain length extending radially outwards from this fixed point(it is attached to the fixed point). To the free end of the fixed rod, an object is present and it is capable of changing it's speed(by thruster say or any convenient method. And ignore any resistance). It starts with a certain speed but say it's speed continuously increases as it goes around. Now, I initially thought that the orbit would not be circular but then I couldn't explain why. My friend gave the reasoning that the rod is rigid and of a fixed length so it would be a circular path no matter what. So here's the first clarification I need..... even theoretically, a body cannot be perfectly rigid right? This would mean influences would travel through the body instantaneously right?

Second, I searched up online and I found, regardless of whether the speed is increasing... the object can still maintain a circular path. But i have a doubt....

In the example I gave(and assuming I am correct about the fact that no body is perfectly rigid) the changing speed would also increase the value of the centripetal force right(Centripetal Force∝v²). Wouldn't this compress(even if very slight) the rod? If that's the case, even normally when there is a constant centripetal force, that too would compress the rod at least a tiny bit right.

I understand that this may be negligible and my friend's answer and what I found online can be a sufficient answer. Hence I just have 2 questions:

1.No matter how tiny the decrease in length is(Assuming my reasoning is correct), in the end it still ends up deviating from a perfect circular path. So technically, would I be correct(or is it fine to approximate it to be a perfect circle)?
2.If it is negligible for small speeds, are there speeds the object can possess such that these changes are not negligible?

1.The scenario

2.Images from Wikipedia, [AI reference redacted by the Mentors] and Physics Stack Exchange respectively.

 

[PeroK]

TL;DR Summary: I was wondering about whether non uniform circular motion can exist?

Motion of an object in a perfect geometric shape, such as a circle, is part of the mathematical theory. Which will never model a physical scenario perfectly. An object is not a point and a trajectory can never be a perfect circle - even before quantum mechanical considerations enter the theoretical picture.

 

[A.T.]

1.No matter how tiny the decrease in length is(Assuming my reasoning is correct), in the end it still ends up deviating from a perfect circular path

There are no perfect circular paths in nature. Even a passive object without the thrusters will not move on a perfect circle. So making your objection specific to non-uniform circular motion makes no sense.

You don't provide any rationale why non-uniform circular motion should be impossible (in the idealized case), in contrast to uniform circular motion.

Your reasoning based on purely tangential thrust and all the perpendicular force acting via a non-rigid rod is just a special case which doesn't prove that something is impossible. You could just as well provide the additionally required centripetal force via radial thrusters. Or even provide all centripetal force via thrusters and get rid of the rod.

 

[Lnewqban]

In the example I gave(and assuming I am correct about the fact that no body is perfectly rigid) the changing speed would also increase the value of the centripetal force right(Centripetal Force∝v²). Wouldn't this compress(even if very slight) the rod? If that's the case, even normally when there is a constant centripetal force, that too would compress the rod at least a tiny bit right.

The centrifugal effect should induce a traction effect on the road.
Therefore, increasing the rotational velocity of the object should elongate the road rather than compress it, as long as the fixed point is able to stand the rotating radial force.
Try a quick experiment using a rubber band instead of a rod.

 

[A.T.]

increasing the rotational velocity of the object should elongate the road rather than compress it,

Yeah, I did not understand that part of the question either. Since it's about any deviation from circular it doesn't really matter, but potentially indicates some more fundamental misconceptions.