I know that mechanical watches have a spring that they wind to store energy, and un-winding the spring produces energy for the watch. But a spring produces a lot of force when it’s very tightly wound, and very little when it’s almost completely un-wound. So how does the watch even that out with high precision?
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You have the main spring which basically just keeps everything tensioned up. Then if it’s one of those automatic watches. There’s a mechanism that is off balance and spins around with your movements. This spinning around can turn the main spring mechanism.
The propulsive force is the *mainspring*. This is what powers the watch, and yes, the force does change as it unwinds, but it’s designed to maintain more force than necessary as it unwinds over the course of 30-40 hours.
The mainspring is regulated by the *escapement* which is a gearing mechanism that only lets the mainspring unwind in small little periodic increments. This is the ticking sound that you hear in a watch.
The escapement is regulated by the *balance wheel* which houses the *hairspring* which is a tightly wound coil that winds and unwinds. The force of the mainspring puts a thrust on the escapement, which stops the mainspring for a moment. But that trust is transmitted into the hairspring, which winds and then unwinds at a high frequency (usually 3-5 beats per second).
Once the hairspring is unwound, it releases the escapement, which lets the mainspring unwind one more increment, and so on. The mainspring is what powers the geartriain that moves the hands.
The trick is to select reasonably temperature insensitive materials for the hairspring, which is constantly winding and unwinding as the “heartbeat” of the watch. More expensive watches have hairsprings set to run at faster frequencies for more accuracy, and they are fine-tune in more ways to counter position against gravity, temperature and magnetism.
So to answer OP’s question, yes, spring unwinding force is not constant, but using a separate spring that winds and unwinds in one full cycle counteracts that, letting you create a part with consistent periodic motion.
In short, the accuracy gets worse the more the mainspring winds down. There are some types of constant force escapements or drive systems that fight this but on rare more expensive watches.
For example in a chain drive watch (fusee & chain) the chain gets unwrapped and wrapped on sprockets of varying diameters similar to a CVT gearbox which equalizes the torque.
There are plenty of great technical answers here, but how about an analogy? As others have pointed out, mechanical watches have a device inside that acts as a pendulum. Picture pushing a kid on a swing. The rate of swinging is the same whether you give the kid a gentle push or a more forceful one. As long as you’re pushing at all, the swing keeps swinging (at different heights maybe) at the same rate.
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