A solid uniform sphere of mass m and radius R rolls without...

A solid uniform sphere of mass $m$ and radius $R$ rolls without slipping down an incline of height $h$. (a) What forms of mechanical energy are associated with the sphere at any point along the incline when its angular speed is $\omega$ ? Answer in words and symbolically in terms of the quantities $m, g, y, I, \omega$, and $v$. (b) What force acting on the sphere causes it to roll rather than slip down the incline? (c) Determine the ratio of the sphere's rotational kinetic energy to its total kinetic energy at any instant.

Key Concepts

Mechanical Energy

Mechanical energy in a system is the sum of its potential energy and kinetic energy. In rolling motion, kinetic energy has two parts: the translational kinetic energy due to the motion of the center of mass and the rotational kinetic energy resulting from the object spinning about its axis. Understanding this decomposition is essential for analyzing energy conversion in systems moving under gravity.

Rolling Without Slipping

Rolling without slipping is a condition where there is a fixed relationship between an object's translational velocity and its angular velocity (v = ?R). This constraint prevents the point of contact from sliding on the surface, ensuring that friction acts to enforce rolling rather than to dissipate energy through slipping.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion, depending on how its mass is distributed relative to the axis of rotation. It appears in expressions for rotational kinetic energy and plays a vital role in determining how easily an object can be accelerated rotationally.

Static Friction

Static friction is the force that prevents the relative sliding of surfaces in contact. In the context of rolling, static friction provides the necessary torque to initiate and sustain rotation without slipping, despite not doing work in the idealized non-dissipative rolling scenario.

Energy Conservation

The principle of energy conservation states that in the absence of non-conservative forces, the total mechanical energy of a system remains constant. This principle is used to relate the gravitational potential energy lost as the object descends an incline to the gain in both translational and rotational kinetic energies.

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