Understanding constant speed props
I'm doing some reading to about constant speed props, prior to using one. One article 'Pelicans Perch: Those Marvelous Props' is frequently recommended so I'm starting there. It raises a question which I'm hoping someone can answer.
The article starts by taking about the different props used to optimise different performance (take off, climb, cruise etc). One statement says: "Some props used counterweights to balance aerodynamic forces so that if the prop were under a load, the blades would flatten out automatically, increasing RPM and power." I understand why the RPM would increase but I don't understand why the power would increase. Applying my school boy physics, I thought power would be a function of the pitch and the RPM. So, if the RPM increases but the pitch flattens out, the net power would still be the same. Where am I going wrong? And why does a flatter blade work any better for the take off than it does, say, for a climb or cruise? I seem to be missing some part if the jigsaw here.