What is the moment of inertia of an object that rolls...

What is the moment of inertia of an object that rolls without slipping down a 2.00-m-high incline starting from rest, and has a final velocity of 6.00 $\mathrm{m} / \mathrm{s}$ ? Express the moment of inertia as a multiple of $M R^2$, where $M$ is the mass of the object and $R$ is its radius.

Key Concepts

Translational and Rotational Kinetic Energy

These are the two forms of kinetic energy in rolling motions. Translational kinetic energy is associated with the movement of the object's center of mass and is given by (1/2)Mv², while rotational kinetic energy is associated with the object's rotation about its center of mass and is given by (1/2)I?². The combination of these energies under the conservation of energy framework allows one to solve for unknowns like the moment of inertia or final velocity.

Conservation of Energy

This principle is used to equate the total mechanical energy at different states of the motion, particularly when there is no energy loss due to friction that does work (static friction does no work during rolling without slipping). In the context of rolling down an incline, the gravitational potential energy is converted into both translational and rotational kinetic energy.

Moment of Inertia

This is a measure of an object's resistance to rotation about an axis. It depends not only on the mass of the object but also on how that mass is distributed relative to the axis of rotation. In problems of rolling objects, the moment of inertia often appears in combined rotational and translational dynamics, and it is typically expressed in terms of the object's mass and geometry.

Rolling Without Slipping Condition

This is the constraint that relates the linear motion of the center of mass to the rotational motion of the object. Under this condition, there is a fixed relationship between the translational velocity (v) and the angular velocity (?) given by v = ?R, where R is the radius of the object. This condition ensures that the point of contact between the rolling object and the surface does not slide, which is critical for applying energy conservation accurately.

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