If a tail dragger is converted to a tricycle gear airplane, what will be the effect on weight distribution?
There are some planes out there that exist in both configurations. One of the more common and first that comes to mind is the popular Cessna 150, the Cessna 172 also has an available conversion. 

As a taildragger


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And with tricycle gear 

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Since the majority of the mass remains unmoved (the engine, fuel tanks etc) the CG will see a bit of a shift but will not be that drastic. The biggest weight shift will be moving the mains up front. However since your gear is located in a different place and thus is loaded differently your CG limits may change.