A sphere of mass M and radius R is in pure rolling. Find the force exerted by upper hemisphere on lower hemisphere.

To determine the force exerted by the upper hemisphere on the lower hemisphere of a sphere in pure rolling motion, we need to analyze the forces and the distribution of weight within the sphere. Let's break this down step by step.

Understanding the Forces at Play

When a sphere rolls without slipping, it experiences both translational and rotational motion. The center of mass of the sphere moves in a straight line, while the sphere rotates about its center. The key forces acting on the sphere include:

• The gravitational force acting downward, equal to the weight of the sphere (Mg).
• The normal forces exerted by the ground and the contact forces between the upper and lower hemispheres.

Weight Distribution in the Sphere

The weight of the sphere is uniformly distributed throughout its volume. However, when considering the interaction between the upper and lower hemispheres, we need to focus on how the weight is shared between them. Since the sphere is symmetrical, we can infer that the weight is evenly distributed between the two hemispheres when the sphere is at rest. However, in rolling motion, the dynamics change slightly due to the rotation.

Calculating the Force Exerted by the Upper Hemisphere

In pure rolling motion, the center of mass of the sphere is accelerating, which affects the forces acting on each hemisphere. The force exerted by the upper hemisphere on the lower hemisphere can be understood through the concept of normal force.

Let’s denote:

• F: the force exerted by the upper hemisphere on the lower hemisphere.
• Mg: the total weight of the sphere.

In a static scenario, each hemisphere would support half of the weight of the sphere:

F_static = Mg/2

However, in rolling motion, the dynamics change slightly due to the acceleration of the center of mass. The force exerted by the upper hemisphere must not only support its own weight but also account for the inertia of the lower hemisphere as the sphere rolls. The effective force can be calculated using the following reasoning:

Dynamic Considerations

When the sphere rolls, the lower hemisphere experiences a downward force due to gravity and an upward force due to the contact with the ground. The upper hemisphere, in turn, exerts a force on the lower hemisphere to maintain equilibrium. The net force acting on the lower hemisphere can be expressed as:

F = Mg/2 + (M/2)(a/g)

Where a is the linear acceleration of the center of mass and g is the acceleration due to gravity. In pure rolling motion, the relationship between linear acceleration and angular acceleration is given by:

a = Rα

Where α is the angular acceleration. Thus, we can substitute this into our equation for F:

F = Mg/2 + (M/2)(Rα/g)

Final Expression for the Force

To summarize, the force exerted by the upper hemisphere on the lower hemisphere during pure rolling can be expressed as:

F = Mg/2 + (M/2)(Rα/g)

This equation shows that the force is influenced by both the gravitational force acting on the sphere and the additional force due to the rolling motion. The exact value of α can be determined based on the specific conditions of the rolling motion, such as the radius of the sphere and the surface it rolls on.

In essence, the interaction between the two hemispheres is a fascinating example of how forces distribute in a dynamic system, illustrating the principles of mechanics in action.