I have a hard time understanding the physics behind the pneumatic erecting system in an attitude indicator. According to my book and the linked image, the applied force is “precessed 90 degrees” (anti-clockwise seen from above) and thus re-aligns the gyro. I understand how the pendulous vanes change the exhaust of the air, but since this part is stationary, i.e. not rotating, how can the net force be precessed?
Gyroscopes have two types of motion.
The gyroscope spins within its housing (usually at a pretty high speed). The housing can change attitude within some limits.
The vanes aren't part of spinning gyroscope, but they are part of the housing (which can shift/rotate with respect to the airframe). The vanes create a torque to change the attitude of the housing.
There is a youtube video with some animations that try to show this effect.