Methods and apparatus for weighing an article, such as a mail piece, while the article is moving at high speed, for example along a transport of a sorter machine. An article (900) is received from an intake transport (1200), and gripped in a weighing station (1310), in between a capstan roller and a pinch roller (1316), which are synchronized to minimize slipping. A first precision servo system (1252, 1250) alters the speed of the article, and in the process acquires torque data for storage and analysis (1212, 1282). A second precision servo system (1260,1330) applies a constant force, via a tension arm (1320), urging the pinch roller (1316) against the capstan roller, independently of the thickness of the mail piece. Active electronic damping (FIG. 19) preferably is applied in the second servo system minimize vibration of the tension arm, but only while gripping the article.