A basketball of mass 610 g and circumference 76 cm is rolling without slipping across a gymnas

step 1 This problem will cover rotational kinetic energy. Say we have a basketball, which is rolling wi

A basketball of mass 610 g and circumference 76 cm is...

A basketball of mass $610 \mathrm{~g}$ and circumference $76 \mathrm{~cm}$ is rolling without slipping across a gymnasium floor. Treating the ball as a hollow sphere, what fraction of its total kinetic energy is associated with its rotational motion? a) 0.14 b) 0.19 c) 0.29 d) 0.40 e) 0.67

Key Concepts

Rolling Without Slipping

This concept refers to the condition when an object, such as a sphere, rolls in such a way that its point of contact with the surface is momentarily at rest. This means that the translational velocity of the center of mass is directly related to the angular velocity through the relation v = R?, where R is the radius of the sphere. It ensures a direct link between the rotational and translational kinetic energies.

Rotational Kinetic Energy

Rotational kinetic energy is the energy an object possesses due to its rotation and is given by the formula KE_rot = 1/2 I?^2, where I is the moment of inertia and ? is the angular velocity. In the context of rolling motion, this energy contributes alongside translational kinetic energy to the total kinetic energy of the object.

Moment of Inertia for a Hollow Sphere

The moment of inertia quantifies an object's resistance to changes in its rotation and depends on its mass distribution. For a hollow sphere, the moment of inertia is given by I = (2/3) mR^2. This specific formulation is crucial in determining how the mass distribution affects the rotational kinetic energy of the sphere.

Energy Partitioning in Rolling Motion

When an object rolls without slipping, its total kinetic energy is partitioned into translational kinetic energy (associated with the motion of its center of mass) and rotational kinetic energy (associated with its spin). The ratio of these energies depends on the moment of inertia and the rolling condition, allowing us to calculate the fraction of energy stored in rotation compared to the overall kinetic energy.

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