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When an object like a solid sphere rotates, it possesses a specific kind of kinetic energy due to this motion called rotational kinetic energy. This energy is analogous to the kinetic energy of an object that moves in a linear path but applies to rotation.

For an object with a moment of inertia, represented by the symbol 'I', and an angular speed 'ω' (omega), the expression for the rotational kinetic energy is given by the formula: \[\frac{1}{2}I\omega^2\].

In the context of the exercise involving a sphere descending an incline, we apply this concept to understand that as the sphere gains speed down the hill, the energy associated with its spinning - or rotational kinetic energy - increases. Since the sphere rolls without slipping, we also know that there is a direct relationship between its rotational speed and its linear speed. Understanding the distribution of kinetic energy between translation (linear motion) and rotation is fundamental in physics, particularly when analyzing rolling objects.

Potential energy is the energy stored by an object due to its position relative to a force. In physics, when we consider a height 'h' in a gravitational field represented by the acceleration due to gravity 'g', the potential energy, often gravitational potential energy, is given by the equation: \[mgh\].

In our textbook exercise, the sphere starts at a certain height on an incline, indicating that it has potential energy at the beginning of its motion. As the sphere rolls down the incline, this potential energy is converted into kinetic energy — both linear and rotational. The potential energy directly relates to how far an object can fall under the influence of gravity, and in this case, 'y' represents the vertical distance above the base of the incline. This potential-to-kinetic energy transformation is a vivid illustration of the conservation of mechanical energy in action.

Static friction force is the force that resists the start of sliding motion between two surfaces that are in contact. It is the frictional force that acts when an object is at rest or when there is no relative motion between the surfaces.

In our sphere on an incline scenario, the static friction force is crucial because it prevents slipping; it allows the sphere to roll, matching linear and angular accelerations. It does this by applying a torque at the point of contact with the surface, which results in the sphere spinning rather than just sliding down. Essentially, static friction is what enables the rotational kinetic energy to manifest in rolling objects be it wheels, balls, or cylinders, making it intrinsic to understanding rolling motion in physics.