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Kinetic energy is the energy that an object possesses due to its motion. In rotational kinematics, this energy is divided into two components: translational kinetic energy and rotational kinetic energy.

Translational kinetic energy is associated with the linear motion of the center of mass. It is calculated using the formula:

• \( KE_{\text{translational}} = \frac{1}{2} m v^2 \)

Here, \( m \) is the mass and \( v \) is the velocity of the center of mass. Rotational kinetic energy, on the other hand, relates to the object's rotation about its center of mass:

• \( KE_{\text{rotational}} = \frac{1}{2} I \omega^2 \)

Where \( I \) is the moment of inertia and \( \omega \) is the angular velocity. These two forms of kinetic energy come together in rolling objects, where the total kinetic energy is the sum of both types. In the exercise, this relationship allows us to find rotational kinetic energy from the total given energy.

Moment of inertia, often symbolized as \( I \), measures an object's resistance to changes in its rotational motion. It can be thought of as the rotational equivalent of mass in linear motion.

Each shape and distribution of mass has a different formula for calculating the moment of inertia. For a hollow sphere, like the one in the exercise, the moment of inertia is provided and is crucial for computing the rotational kinetic energy.

In general, the moment of inertia depends on:

• The mass of the object
• How the mass is distributed relative to the axis of rotation

A larger moment of inertia means the object is harder to rotate. In the problem, knowing the moment of inertia of the sphere indicates how much of the kinetic energy is rotational versus translational. Because the sphere is rolling, the moment of inertia is used to determine rotational kinetic energy and relate it to the linear speed of the center of mass by applying rotational kinematics principles.

When an object rolls without slipping, there is a specific relationship between its linear and angular velocity. This condition means that the object rolls in such a way that its point of contact with the surface is momentarily at rest relative to the surface.

Mathematically, this is expressed as:

• \( v = r \cdot \omega \)

Where \( r \) is the radius of the object, \( v \) is the linear velocity of the center of mass, and \( \omega \) is the angular velocity. This relationship allows one to calculate \( \omega \) if \( v \) is known, or vice versa.

In the exercise, this principle means that as the sphere rolls up the inclined plane without slipping, both its rotational and translational motion can be described using this single simple relationship. This is useful as it allows us to express rotational properties in terms of translational variables (and vice versa), greatly simplifying the calculation of kinetic energies.

An inclined plane is a flat surface tilted at an angle to the horizontal. It modifies the effects of gravity on an object relative to movement along its surface. The key factor here is the angle of inclination \( \theta \), which influences the component of gravitational force causing the object to accelerate or decelerate along the plane.

When analyzing motion on an incline, we often break down the gravitational force into:

• A component perpendicular to the inclined plane, which doesn't affect the motion
• A component parallel to the plane, which drives motion or resists it

In the exercise, as the hollow sphere moves up the incline, gravitational potential energy changes. This change affects the sphere's total kinetic energy as it rolls. The angle of 30 degrees specifically defines how much gravitational energy is converted to potential energy when the sphere climbs the incline, directly impacting its kinetic energy — a concept underpinned by energy conservation laws.