Problem Set

NBPhO 2026

6. Pull-Up Bar Rope Jaan Kalda 5 pts

Mechanics · Kinematics, Constraints, Statics

A rope pulled at constant rate over a horizontal bar drags a weight across a frictionless floor; find its speed, acceleration, and the angle at which it lifts off.

Problem by Jaan Kalda.

A rope is thrown over a horizontal pull-up bar at height HH above the frictionless floor. One end of the rope is tied to a heavy weight resting on the floor; the other end is pulled so that the length of the rope between the bar and the weight decreases at a constant rate vv. At a given moment, this segment makes an angle α\alpha with the vertical (see figure); the weight is still in contact with the floor. Gravitational acceleration is gg.

Graph: A rope passes over a horizontal pull-up bar at height H; one segment hangs vertically with speed v upward, the other segment makes angle α with the vertical and connects to a heavy weight on a frictionless floor.

i) (1 point) Find the speed uu of the weight.

ii) (1 point) Find the magnitude of the acceleration of the weight.

iii) (3 points) Find the angle α0\alpha_0 at which the weight lifts off the floor.