There is a damper so either the damper (#7) is broken or the hinge pin (#8) has become detached. Part numbers are as follows:
Don't know what year your car is, so provided both versions of part #7
#7 8E2880324 up to 19/05/2014 £36
#7 4H2880324 after 19/05/2014 £32
#8 8E0880802 £1.20
The first minute of the following video will show you how to remove the glovebox:
https://www.youtube.com/watch?v=WHiULh5xKAg
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2001 A8 D2 FL 3.7 Q - LPG Conversion, Engine: AKC, Gearbox:FBB, Colour:LY5X Aqua Blue pearl effect, Interior:WJ light beige/dark beige valcona leather with burr walnut inserts
My DIY jobs -
Replaced alarm horn,
Replaced Coolant Temp sensor,
Updated RNS-D firmware,
Installed Reversing Camera and Digital TV,
Dashcam Installation,
Retrofitted Rear Blinds,
Auto-Dimming Rear View Mirror,
Chrome Boot Struts
Sold! 
2008 S8 D3 5.2 V10 - Engine: BSM, Gearbox:KLW, Colour:LY7F Suzuka Grey Pearl Effect, Interior:Black/Black with Silver/Black valcona leather with carbon atlas inserts, Bang & Olufsen, Heated windscreen, Soft-close doors, TV, DAB, AMI, Quad Zone, Auto Boot, Auto dim Mirrors, Keyless entry, PDC with rearview camera, AFS II Headlights
My DIY jobs -
Fan Jet Windscreen Washers,
Retrofitted Electric Side Blinds