1. Two cylinders A and B of diameters r1 and r2 weighing M both respectively are placed as shown in the figure. Find the reaction forces at the walls assuming all contact surfaces are frictionless

To determine the reaction forces at the walls for the two cylinders A and B, we need to analyze the forces acting on each cylinder due to their weights and the contact with the walls. Since the surfaces are frictionless, the only forces we need to consider are the normal forces exerted by the walls on the cylinders and the gravitational forces acting on them.

Understanding the Setup

Let’s visualize the scenario. We have two cylinders, A and B, both with the same weight M. The diameters of the cylinders are r1 and r2, respectively. When placed against the walls, each cylinder will exert a force on the wall due to its weight, and the wall will exert an equal and opposite reaction force on the cylinder, according to Newton's third law.

Analyzing Forces on Each Cylinder

For each cylinder, the forces can be broken down as follows:

• Weight (W): Each cylinder has a weight acting downwards, which is equal to M (the mass) multiplied by g (acceleration due to gravity).
• Normal Force (N): This is the force exerted by the wall on the cylinder, acting perpendicular to the wall.

Since the surfaces are frictionless, the only vertical force acting on each cylinder is its weight, and the normal force from the wall must balance this weight to maintain equilibrium.

Equations of Equilibrium

For cylinder A, the forces can be expressed as:

• Vertical forces: N_A = W_A = M * g

Similarly, for cylinder B:

• Vertical forces: N_B = W_B = M * g

Calculating the Reaction Forces

Now, since both cylinders are in equilibrium, the normal forces exerted by the walls on each cylinder will be equal to the weight of the respective cylinder:

• For cylinder A: N_A = M * g
• For cylinder B: N_B = M * g

Thus, the reaction forces at the walls can be summarized as:

• Reaction force at the wall for cylinder A: N_A = M * g
• Reaction force at the wall for cylinder B: N_B = M * g

Final Thoughts

In conclusion, the reaction forces at the walls for both cylinders A and B are equal to their respective weights, which is M * g. This analysis assumes that the cylinders are not moving and that the surfaces are perfectly frictionless, allowing us to focus solely on the balance of vertical forces. If there were any additional forces or if the cylinders were not in equilibrium, we would need to consider those factors as well.