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Question about Grounding

Hi,
In the attached picture, should I connect the GND-1 in Arduino to the GND-2 (or the negative side) of the VDC power supply ?
If the answer is Yes, then what will happen if GND-1 is not connected to GND-2 ?

Thanks for your help in advance

Regards
Jack
 

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Deja vu...????
Could swear I've seen this query previously.
Then, as now, the answer is absolutely, otherwise there is no return path for the signal to the mosfet gate.
Signal is floating and will create erratic behaviour at best, nothing at worst.
 
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