This is late but:
It might help to know what equipment you've been given.
But honestly, I have no idea how to approach this. Obviously you want to keep v^2/r constant, but I suppose the question is how can you best do it, so as to minimise the error in your results. Especially since it's telling you to keep it constant "even when radius is changing".
My first thought was to swing it in a vertical arc, and then use gravity to kind of 'calibrate' your v^2/r value, but that really presents more problems than solutions, especially if you're swinging your mass by hand.
I suppose I'll have to go with what everyone else is saying and just say "keep the period and radius constant" even though it's not really answering the question of how to keep v^2/r constant with a changing radius.