Strength at the bottom is the indexing radical. Affairs and duties are the things one applies effort to, so 力 grounds 务 in the family of exertion words - 努, 助, 劳 - where physical or institutional force is brought to bear.
The down-pointing foot on top is a stylised remnant of 攵 (a striking action) in the traditional 務. After simplification from 務 in 1956, it survives as an abstract cap suggesting effort directed downward onto the work below.
