A 620 -g basketball rolls without slipping across a level...

A 620 -g basketball rolls without slipping across a level floor at a speed of $1.0 \mathrm{~m} / \mathrm{s}$. At what speed would a 40.0-g glass marble travel, as it rolls without slipping, if it has the same kinetic energy as the basketball? Model the basketball as a thin spherical shell and the marble as a uniform solid sphere. Example 8-10

Key Concepts

Rolling Without Slipping

This concept refers to an object that is both translating and rotating in such a way that the point of contact with the surface has zero velocity relative to that surface. It establishes a relationship between translational speed and angular speed, typically expressed as v = R?, where v is the linear speed, ? is the angular speed, and R is the radius of the object.

Rotational Kinetic Energy

Rotating objects in motion possess both translational and rotational kinetic energy. The total kinetic energy is therefore the sum of the translational kinetic energy, (1/2)mv², and the rotational kinetic energy, (1/2)I?². Under the rolling without slipping condition, the angular speed can be substituted with the linear speed, linking the two energy components.

Moment of Inertia

The moment of inertia is a measure of how the mass of an object is distributed with respect to its axis of rotation, affecting the object’s resistance to changes in its rotational motion. Different objects (or models, such as a thin spherical shell versus a solid sphere) have different moments of inertia, which significantly influence the rotational kinetic energy component during rolling motion.

Energy Equivalence in Rolling Objects

This involves equating the total kinetic energy of different rolling objects to compare their speeds or other dynamic quantities. By using the expression for total kinetic energy in rolling motion and the respective moments of inertia for each object, one can derive relationships or compute the speeds necessary for energy equivalence between bodies of different masses and mass distributions.

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