To solve the problem of the normal reaction between the top sphere and the hemispheres below it, we need to analyze the forces acting on the system. The setup involves three solid hemispheres of radius R, placed in contact with each other on a rough horizontal surface, and a sphere of mass m and radius R placed symmetrically on top of them. Let's break down the situation step by step.
Understanding the Forces at Play
In this equilibrium scenario, we have several forces to consider:
- The weight of the top sphere, which acts downwards and is equal to mg, where g is the acceleration due to gravity.
- The normal reactions from the hemispheres, which act upwards against the weight of the top sphere.
Weight Distribution
Since the top sphere is symmetrically placed, the weight is evenly distributed among the three hemispheres. Therefore, each hemisphere will experience an equal share of the normal force exerted by the top sphere.
Calculating the Normal Reaction
Let’s denote the normal reaction force from each hemisphere as R. Since there are three hemispheres supporting the top sphere, the total normal force exerted by the hemispheres combined must equal the weight of the top sphere:
3R = mg
From this equation, we can solve for the normal reaction force R:
R = mg / 3
Conclusion on Normal Reaction
Thus, the normal reaction between the top sphere and each hemisphere, assuming the system is in equilibrium, is given by:
R = mg / 3
This means that each hemisphere supports one-third of the weight of the top sphere. This distribution is crucial for maintaining equilibrium and ensuring that the system remains stable.
Visualizing the Setup
To further clarify, imagine the three hemispheres forming a triangular base with their flat faces on the ground. The top sphere sits perfectly balanced on this triangular arrangement. The forces acting on it are balanced, which is why we can confidently say that the normal reaction is evenly distributed among the three supporting hemispheres.
In summary, the normal reaction force from each hemisphere is one-third of the total weight of the top sphere, ensuring that the entire system remains in a state of equilibrium.
