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In a glider, sometimes it's useful to do a side slip with crossed controls. However, when that happens, the pitot tube angle is out of line and the ASI goes to zero, therefore losing any indication of the current airspeed until the sideslip is interrupted.

During this maneuver, what would happen if I were not to keep proper control of the attitude, and thus airspeed, and were to stall the wings? would the glider spin, or would the unusual asset prevent it?

Yes if you were to stall the airplane while in a side slip it will probably depart, or start to depart, into a spin. How aggressively would depend on the glider's actual stall behavior, some being more benign than others, but say you were in a glider with an abrupt stall with minimal warning and a sharp nose drop. That sort of glider would spin if you didn't initiate recovery right away.

This isn't such a big problem, because the nice thing about gliders is you don't have power to confuse things when it comes to pitch attitude vs speed. For a given configuraiton (weight/CG etc), if attitude X gives speed Y than you can be certain that any time you pitch to attitude to X you will get speed Y, with or without an airspeed indicator.

So, if you are on final at 50 kt, and want to side slip, just make sure you lower the nose a couple of degrees during the slip and control the pitch attitude, so that you will maintain somewhere around 50kt with the additional drag and there is very little risk of stalling and spinning. Easy peasy.

You also have the other impending stall cues, like softening controls, noise dying off, and such, so if you are aware and on top of things there shouldn't be a problem and it would take some ham fisted flying to really mess up.

If you are getting trained well, there should be dedicated no-airspeed instruction and practice and once you are considered a proficient pilot you should have no problem with flying safely with no airspeed for all maneuvers (if you sent me out in a power plane on some kind of wacky challenge, and told me I could only have one operating flight instrument, my choice would be a compass).

Firstly, I would find a way to keep that airspeed info coming. Loss of airspeed info certainly could be very dangerous, especially if you are slipping into your landing approach in any kind of winds.

That being said, if you are cross controlled (ailerons in rudder away), as soon as you feel a buffet center the ailerons and rudder, and PITCH DOWN. If you haven't started yawing, you are OK. If you have, use that rudder!

Breaking/preventing the stall by pitching down is critical, as is controlling yaw with the rudder. Do not worry about your bank, you can carefully roll out after the stall is definitely broken. Only then can you safely use aileron inputs.

Provided proper inputs (especially centering ailerons) are done at stall warning, spin should not develop if the CG is set up correctly and the glider is of a stable design.
I would worry a lot more about my airspeed indicator.

Remember, the best way to know the consequences of a stall is to try it with a qualified instructor at a safe altitude. This is the best way to learn the slow flight characteristics of your plane.

If your design is docile enough, simply relaxing the elevator may be enough when you feel the buffet, but practicing up high will help fine tune your technique tremendously.