Sand on moving conveyor belt · CAP 2006 #Problem 1
Physics · mechanics
Problem
Sand from a stationary hopper falls onto a moving conveyor belt at a rate of 5.00 kg/s as shown above. The conveyor belt is supported by frictionless rollers and moves at a constant speed of 0.75 m/s under the action of a constant horizontal external force F_ext supplied by the motor that drives the belt. [Figure: Diagram showing sand falling from hopper onto moving conveyor belt with external force F_ext] Find: 1) the force of friction exerted by the belt on the sand; 2) the external force F_ext; 3) the work done by F_ext in 1 s; 4) the kinetic energy acquired by the falling sand each second due to the change in its horizontal motion; 5) compare and analyse the answers to 3) and 4).Intuitive explanation
This problem explores the subtle difference between the work done by an external force and the kinetic energy gained by the system—a clash between energy accounting and momentum transfer. Sand arrives with zero horizontal velocity and must be accelerated to belt speed by friction; that friction force does work on the sand, but the motor must do more work because it also fights the reaction force from accelerating the sand. The key insight is recognizing that some of the motor's work goes into kinetic energy of the sand, while the rest dissipates as heat due to sliding friction between sand and belt.
Intuitive explanation
This problem explores the subtle difference between the work done by an external force and the kinetic energy gained by the system—a clash between energy accounting and momentum transfer. Sand arrives with zero horizontal velocity and must be accelerated to belt speed by friction; that friction force does work on the sand, but the motor must do more work because it also fights the reaction force from accelerating the sand. The key insight is recognizing that some of the motor's work goes into kinetic energy of the sand, while the rest dissipates as heat due to sliding friction between sand and belt.