A disk of mass m on a frictionless table is attached to a...

A disk of mass $m$ on a frictionless table is attached to a hanging cylinder of mass $M$ by a cord through a hole in the table (see Fig. 5-42). Find the speed with which the disk must move in a circle of radius $r$ for the cylinder to stay at rest.

Key Concepts

Newton's Second Law of Motion

Newton's Second Law of Motion states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This principle is used to relate the tension (which provides acceleration in circular motion) to the gravitational force acting on the hanging mass, allowing for the determination of the necessary speed of the disk.

Force Equilibrium

Force equilibrium refers to the state where the sum of all forces acting on an object is zero, resulting in no acceleration. For the hanging mass to remain stationary, the upward tension in the cord must exactly balance the downward gravitational force acting on it.

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path, directed toward the center of the circle. It is calculated as the product of the mass of the object and the square of its velocity divided by the radius of the circle, and it explains why an object constantly changes direction when traveling along a curved path.

Tension in the Cord

Tension is the force transmitted through a rope, string, or cord when it is pulled tight by forces acting from opposite ends. In this context, the same tension force is responsible for providing the centripetal force needed for the disk's circular motion and for balancing the weight of the hanging mass, ensuring that the hanging mass remains at rest.

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