Problem 37 - An Introduction to the Mechanics of Solids in SI Units

Question

A circular cylinder \(A\) rests on top of two half-circular cylinders \(B\) and \(C\), all having the same radius \(r\) The weight of \(A\) is \(W\) and that of \(B\) and \(C\) is \(1 / 2 W\) each. Assume that the coefficient of friction between the flat surfaces of the halfcylinders and the horizontal table top is \(f\). Determine the maximum distance \(d\) between the centers of the half-cylinders to maintain equilibrium.

Step-by-Step Solution

Answer

The maximum distance \(d\) between the centers of the half-cylinders to maintain equilibrium is \(\frac{4r}{3}\)

Step 1: Find the forces acting on system

Determine the forces acting on the system. The weight of the cylinder \(A\) denoted as \(W\) acts downward at the midpoint between cylinder \(B\) and \(C\). Weights of cylinder \(B\), \(\frac{1}{2}W\), and cylinder \(C\), \(\frac{1}{2}W\), also act downward from their respective centers of mass. Because the system is in equilibrium, the sum of these forces equals the frictional forces \(F_B\) and \(F_C\) acting upward on cylinders \(B\) and \(C\) respectively. So we have, \(W + \frac{1}{2}W + \frac{1}{2}W = F_B + F_C\)

Step 2: Sum the torques

Next, consider the torques about the line of action of the frictional force of cylinder \(B\). The sum of these torques should be zero as the system is in equilibrium. The torques caused by weights of cylinder \(A\), \(B\), and \(C\) are \(W\cdot d / 2\), \(-\frac{1}{2}W \cdot r\) and \(-\frac{1}{2}W \cdot (d+r)\) respectively. Therefore, the sum of torques equals 0 gives \(-W \cdot d / 2 - \frac{1}{2}W \cdot r - \frac{1}{2}W \cdot (d+r) = 0\)

Step 3: Solve for \(d\)

Solve the equation obtained in step 2 for \(d\). Rearrange the equation to isolate \(d\), ultimately ending up with \(d = \frac{4r}{3}\)

Key Concepts

Static Equilibrium

Static equilibrium is a state where an object is at rest and the sum of all forces and torques acting upon it equals zero. This concept is essential in understanding why objects like our circular cylinders remain balanced without any movement.
To determine static equilibrium, the following conditions must be met:

• The sum of all horizontal forces must be zero.
• The sum of all vertical forces must be zero.
• The sum of all torques (moments) about any axis must be zero.

In the exercise with cylinders A, B, and C, static equilibrium ensures that the weight forces acting on the system, including those exerted by gravity, are perfectly balanced by the normal and frictional forces. This balance is crucial in determining the maximum distance between the centers of cylinders B and C, as it prevents any rotational or translational movement. Knowing how to apply static equilibrium principles can help you analyze various mechanical problems more effectively.

Friction

Friction is the resisting force that occurs when two surfaces slide against each other. In our case, the friction plays a critical role in maintaining the position of the cylinders, especially on a flat surface like the tabletop. The coefficient of friction, denoted as \( f \), helps to quantify this resisting force.
Cylinder A resting on B and C introduces friction between their surfaces, as well as against the table. The force of friction (\( F \)) can be calculated using the formula:\[F = f \cdot N\]where \( N \) is the normal force. For static equilibrium, the frictional force must be equal and opposite to any applied forces attempting to move the cylinders, thereby preventing motion. Understanding the balance of frictional forces clarifies why and how the spacing between the cylinder centers influences the overall system's stability.

Circular Cylinder

A circular cylinder is a three-dimensional geometric shape commonly found in everyday applications. In our exercise, the solid circular cylinder A rests on the half-circular cylinders B and C, making the concepts of geometry and weight distribution important.
The cylinder's properties include its radius (\( r \)) and its capacity to exert a weight (\( W \)) vertically at its center of gravity. The radius is particularly crucial when considering torque calculations, as it affects the distance components in such equations. For equilibrium maintenance, both geometry and distribution of forces need careful analysis.
Understanding the relationship between the shape of the cylinder and its interaction with other bodies offers insights into why certain alignments, like the maximum distance \( d = \frac{4r}{3} \), ensure that equilibrium is achieved. When studying circular cylinders, being mindful of how dimensions and weights interact leads to a better grasp of mechanical stability.