You have a few remaining options. Air, electricity, and alternative explosives.
A pneumatic weapon uses compressed air or steam to accelerate a projectile. There are rapid firing potato versions, so I don't think you'd have trouble making one that fires bullets instead of spuds. However, the power will be difficult to get up to gunpowder replacement levels. Steam will be easier to use than compressed air, but may not be as easy to store.
A gauss gun works by triggering successive electromagnets to pull a projectile along the barrel. You can actually get a higher energy output than a gunpowder weapon, but the repeat rate may be lower if the power output of the generator is not high enough to recharge the capacitors quickly enough. However, you could simply use many interchangeable capacitors banks or a massive generator.
A railgun is similar to a gauss gun except the projectile is part of the circuit for generating the electromagnetic force. This reduces timing complexities but increases the required precision of the gun and projectiles. As with the gauss gun, higher power output than gunpowder is possible and it will require significant energy storage to fire rapidly.
There are a lot of explosives other than gunpowder. You can use lots of different types of high-explosives to launch projectiles. Presumably, you don't want this though, so I won't expand much.
As for your question on using centrifugal force. With that clever naming gun people are known for, there is the centrifugal gun. It can be hand-cranked or powered by compressed air/steam.
It has been tested several times in the US and billed as the "silent machine gun" but has always been abandoned due to extremely poor accuracy. This makes sense, with centrifugal force the projectile can't be accelerated while it's being guided down a barrel pointed at the target. Either the power is reduced while sliding down a barrel or the accuracy is all in the milliseconds that the projectile is released from the spinning accelerator.
There is a patent on a centrifugal gun from 2003 called DREAD which claims to launch projectiles with the speed of a handgun (300 m/s).