Savchenko Problem 2.7.18
Physics · mechanics · thermodynamics
Problem
A solid disk is tightly mounted on a roller of radius r. Moment of inertia of this system relative to the J axis, mass m. Two threads are wound symmetrically on the roller, on which the system is suspended from a fixed tripod. The threads are vertical. The system is released. Find the acceleration of the disk axis and the tension force of the threads.Intuitive explanation
This is a classic Atwood-machine-like problem where gravity pulls the system down, but the tension in the supporting threads resists that motion. The key insight is to treat this as a rigid body with both translational and rotational motion: the disk accelerates downward while the threads unwind, causing rotation. You'll need two equations—Newton's second law for the center of mass (relating weight, tension, and downward acceleration) and the rotational equation (relating torque from thread tension to angular acceleration)—then use the constraint that linear and angular motion are linked by the roller radius .
Intuitive explanation
This is a classic Atwood-machine-like problem where gravity pulls the system down, but the tension in the supporting threads resists that motion. The key insight is to treat this as a rigid body with both translational and rotational motion: the disk accelerates downward while the threads unwind, causing rotation. You'll need two equations—Newton's second law for the center of mass (relating weight, tension, and downward acceleration) and the rotational equation (relating torque from thread tension to angular acceleration)—then use the constraint that linear and angular motion are linked by the roller radius .