There are already several excellent answers to this question, but I think there's a bit of detail missing from all of them. The fact that rotation stretches a spring doesn't necessarily tell us everything we need to know, as Timaeus suggests. (For my reasoning, see below.) And shouldn't it be possible to show that spinning objects know they are spinning without needing them to have a measurable gravitational field, as answers involving a pendulum require? It seems to me that the final answer must have something to do with basic facts about the local geometry of the universe, and the arrangement of objects in a rotating assemblage.
I am not 100% certain that the below works; I am open to feedback, and I'll edit or delete this answer if it turns out I've made a fundamental mistake. Please let me know!
To get at these issues, I'll consider four scenarios, involving the following experimental apparatus:
- A long, narrow spring of unknown stiffness.
- A set of thrusters attached to the spring, oriented to produce either (perfect) linear or (perfect) rotational acceleration.
- A device attached to the spring that can measure its length when the thrusters aren't firing.
- A memory bank and a simple calculator that can store and perform calculations on those measurements.
Two scenarios add another component:
- A dead-reckoning device that can make continuous measurements of the spring's length during thruster activation.
I won't rely on any precise mathematical formulas, instead paying attention to basic patterns and correlations that we know from experience will certainly hold.
In this scenario, we imagine the spring initially moving linearly, in the direction of its long axis, at some unknown velocity. The thrusters are arranged to produce linear acceleration in either direction. In the first experiment, we do the following:
- Measure the length of the spring ($L_0$).
- Turn on the thrusters for a while in one direction, and then turn them off.
- Measure the length of the spring again ($L_1$).
We can repeat steps 2 and 3, firing in either direction, to get $L_2$, $L_3$, and so on. In this scenario we don't have a dead-reckoner, so we don't know anything about what the spring was doing while the thrusters were firing. And what do we find?
We find that every length measurement is the same—$L_0 = L_1 = L_2$, and so on. So with this set-up, we can learn nothing at all about the spring's linear velocity.
In the second scenario, we make the following changes. Now, in addition to its unknown linear velocity, we imagine the spring rotating about its center of mass at some unknown rotational velocity. The thrusters are arranged to produce clockwise or counterclockwise acceleration around the current axis of rotation, and cause no linear acceleration.
Now we completely ignore linear movement, but follow roughly the same series of steps as above:
- Measure the length of the spring ($L^R_0$).
- Turn on the thrusters for a while in one direction (clockwise or counterclockwise), and turn them off.
- Measure the length of the spring again ($L^R_1$).
Again, we can repeat steps 2 and 3 to get $L^R_2$, $L^R_3$, and so on.
Now things look quite different. Pretty much every time, the length of the spring is different. We don't have any information about what happened while the thrusters fired, but now we can see the change in rotational velocity they caused because every time, the spring has become longer or shorter. Furthermore, we pretty quickly notice that every time they fire in one direction, the spring gets longer, and every time they fire in the other direction, the spring gets shorter.
At this point, it initially appears clear that rotational velocity is somehow different from linear velocity, because one stretches the spring, while the other doesn't. But it's still not totally clear why this should be. And there are still a lot of unknowns. We don't know how fast the spring was moving linearly, in the first case, or rotationally, in the second. And we don't yet seem to be any closer to being able to answer that question in either case. We know that the spring gets longer and shorter in the second case, but what if that's just a kind of "memory effect"? Does it really tell us anything we couldn't know in the first case?
Let's find out! Rather than only measuring the spring's length when acceleration is zero, let's use a dead-reckoner to measure and remember its length during acceleration and see what we can learn.
We return to the linear motion scenario, but we turn on our dead-reckoner, which tracks the length of the spring during thruster activation. We follow the same three steps as before, but taking continuous length measurements. And we find that every time we begin firing a thruster, the spring gets shorter! Then, when we turn the thruster off again, it gets longer again. (Assume that we start and end thruster activations gently, so that the spring doesn't bounce too much.)
Based on this, we have a lot more information. Assuming we know enough about springs and calculus, we can use the length measurements to determine how much force is being generated (at least relative to the unknown stiffness of the spring). If we know the mass of the device as well, we can use that to calculate information about acceleration, and from there, to determine information about how much faster or slower the spring is moving than it was when we first started taking measurements. So even though we don't know how fast the spring was moving at first, we can at least tell the difference between then and now.
After doing this second experiment, the difference between rotational velocity and linear velocity is less obvious. In both Scenarios Two and Three, we can't tell what our initial rotational or linear velocity was. In both scenarios, we can tell when it increases or decreases. In Scenario Two we can tell because the spring gets shorter or longer. In Scenario Three we can tell because we keep track of the forces applied. These are superficially different methods of measurement, but does that superficial difference indicate any deeper difference between these two kinds of velocity? It's harder to tell.
To see the real difference between these two kinds of motion, we have to do one more experiment.
In this experiment, we do the same thing in both the linear and rotational set-ups, with dead-reckoning turned on in the linear case.
- Measure the length of the spring.
- Turn on the thrusters in one direction—call it "to the right" in the linear case, and "clockwise" in the rotational case—for $T$ seconds, keeping track of the velocity change in the linear case.
- Turn off the thrusters and turn them on in the other direction—"to the left" / "counterclockwise"—for $2T$ seconds, keeping track of the velocity change in the linear case.
Now, suppose the first time we do this, we set $T$ to be pretty small, and here's what we see:
- In the linear case, our final change in velocity is to the left.
- In the rotational case, the spring is shorter than it was at first.
Then suppose we repeat step 3, doubling the value of $T$ each time, but continuing to alternate direction between "right" and "left," "clockwise" and "counterclockwise." We quickly see a correlation: each time the final change in linear velocity is to the left, the spring is shorter; each time the final change in linear velocity is to the right, the spring is longer.
Imagine a world in which these correlations held forever. It would be a world in which any spring, when spun around its center of mass in a clockwise direction, stretches. But when spun around its center of mass in a counterclockwise direction, the same spring compresses.
This is clearly not the world we live in. Experience shows that in Scenario Four, eventually, once T gets large enough, the spring will stop compressing when we accelerate it in a counterclockwise direction, and will start stretching again. The only reason it was compressing when we spun it counterclockwise was that at first, it had been spinning at quite a high rate in the clockwise direction, and the resulting centripetal force had stretched it out quite a bit. The point at which the spring stops compressing and starts stretching again is the zero point—the fixed origin against which all other degrees of rotational velocity can be compared. No matter where we are in the universe—no matter what else is nearby—we can always discover that zero point. That's how a spinning object can tell that it's spinning.
Timaeus's answer follows a similar line of reasoning, but imagines that we know a lot of detail about the internal structure of the spring, so that we can tell what length it "should" have. The above shows that we don't even need that knowledge. Any system that behaves roughly like a spring (and many that don't behave like springs at all) will allow us to discover a rotational zero point using the above method.
This series of thought experiments hasn't directly answered the fundamental question—why can objects tell when they are rotating? Instead, it has allowed us to translate it into a different question: Why does spinning a spring faster around its center of mass always cause it to stretch, and never to compress? I'm honestly not sure I can answer that question either, but it sure seems easier!
If I had to venture a guess, it would be this: "because two points inside a circle are always a finite distance apart." You could spin a spring around in a way that would cause it to compress at first—imagine squeezing it very quickly, and spinning it at the same time, so that its ends move closer to the spring's center of mass as it rotates. But eventually, the ends of the spring would stop getting closer, and start getting farther apart again. They would move outside the circle defined by the spring's initial length, and the spring would stretch. The confining force of the spring would then begin to convert the remaining linear momentum into rotational momentum.
I don't know if that's the best way to answer the question. But at the very least, I hope this line of reasoning shows that the best answer relies only on local geometric conditions, and not on anything to do with the state of the rest of the universe.