Two uniform solid balls are rolling without slipping at a...

Two uniform solid balls are rolling without slipping at a constant speed. Ball 1 has twice the diameter, half the mass, and one-third the speed of ball 2 . The kinetic energy of ball 2 is $27.0 \mathrm{~J}$. What is the kinetic energy of ball 1 ?

Key Concepts

Rolling Without Slipping

This concept refers to a motion where an object rolls such that its point of contact with the surface is momentarily at rest relative to the surface. It creates a direct relationship between the translational velocity of the center of mass and the rotational speed about the center. This condition is crucial when analyzing objects that exhibit both linear and rotational motion, as it ensures that energy is distributed between these two forms of motion in a predictable manner.

Translational and Rotational Kinetic Energy

When an object rolls without slipping, its total kinetic energy comprises two parts: translational kinetic energy due to the linear movement of its center of mass and rotational kinetic energy due to its spinning about the axis. Understanding how to calculate and combine these two energy contributions is essential in solving problems in rotational dynamics.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on the mass distribution relative to the axis of rotation. For common shapes like a uniform solid sphere, specific formulas relate the moment of inertia to the object's mass and dimensions. This concept is critical because it determines how rotational kinetic energy changes when the object's size or mass distribution is modified.

Scaling Relationships in Rotational Dynamics

Scaling relationships explore how changes in an object’s dimensions (such as diameter or radius), mass, and speed affect its kinetic energy and moment of inertia. By understanding these relationships, one can determine how adjustments in size or speed influence the overall energy of the system, which is fundamental in comparing different rolling objects under similar conditions.

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