Methods and apparatus for weighing an article, such as a mail piece, while the article is moving at high speed. 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 (1900) reduces oscillation when an inconsistency in thickness of the article is encountered during weighing. The damping force is subtracted from the capstan motor torque data for improved accuracy (FIG. 20B).