Inside a fixed hollow cylinder of radius R whose axis is horizontal two equal cylinders each of radiurs r are placed symmetrically. Athird cylinder identical to each of the cylinders of radius r is placed symmetrically on the smaller two cylinders. Prove that the equilibrium wo'nt exist unless and untill R

To tackle this problem, we need to analyze the conditions under which the system of cylinders can achieve equilibrium. The setup involves a fixed hollow cylinder with a larger radius \( R \) and two smaller cylinders of radius \( r \) placed symmetrically inside it, along with a third identical cylinder resting on top of the two smaller ones. The goal is to prove that equilibrium cannot exist unless a specific condition regarding the radii is met.

Understanding the Geometry of the System

First, let’s visualize the arrangement. We have a large hollow cylinder with radius \( R \). Inside this cylinder, two smaller cylinders, each with radius \( r \), are positioned symmetrically along the horizontal axis. The third cylinder, also of radius \( r \), is placed on top of the two smaller cylinders. For equilibrium to be maintained, the forces acting on the system must balance out.

Analyzing Forces and Stability

In this scenario, the forces acting on the cylinders include gravitational forces and the normal forces exerted by the surfaces they rest on. The gravitational force acts downward on each cylinder, while the normal forces act perpendicular to the surfaces in contact. For the system to be in equilibrium, the following conditions must be satisfied:

• The sum of vertical forces must equal zero.
• The sum of moments about any point must also equal zero.

Condition for Equilibrium

Let’s denote the weight of each smaller cylinder as \( W \). The total weight acting downward on the third cylinder is \( 2W \) (since it rests on two smaller cylinders). For the system to be stable, the center of mass of the entire arrangement must lie within the base of support provided by the two smaller cylinders.

The center of mass of the two smaller cylinders is located at their midpoint, while the center of mass of the third cylinder is directly above this midpoint. The critical factor here is the distance from the center of the larger cylinder to the outer edges of the smaller cylinders. If the distance from the center of the larger cylinder to the edge of the smaller cylinders exceeds \( R \), the system will become unstable.

Mathematical Representation

To express this mathematically, we can state that for equilibrium to exist, the following inequality must hold:

Condition: \( R \geq 2r \)

This means that the radius of the larger cylinder must be at least twice the radius of the smaller cylinders. If \( R < 2r \), the outer edges of the smaller cylinders will extend beyond the confines of the larger cylinder, leading to instability and the inability to maintain equilibrium.

Conclusion

In summary, the equilibrium of this system of cylinders is contingent upon the relationship between the radii \( R \) and \( r \). Specifically, the larger cylinder must have a radius that is at least double that of the smaller cylinders to ensure that the center of mass remains within the base of support. If this condition is not met, the system cannot achieve a stable equilibrium, and the smaller cylinders will not be able to support the weight of the third cylinder effectively.