A composite rolling object is constructed by press-fitting a thin circular ring of…

PHYSICS · Multi-part · CBSE · Physics · System of Particles and Rotational Motion · System of Particles and Rotational Motion

Problem

A composite rolling object is constructed by press-fitting a thin circular ring of mass

M_{r}

and radius

r

concentrically around a solid uniform disk of mass

M_{d}

and radius

R

. The combination rolls without slipping on a horizontal surface with center-of-mass speed

v

. The ring’s radius

r

can be varied while keeping

M_{r}

M_{d}

, and

R

fixed. Assume

r\ge R

. (a) Obtain an expression for the total moment of inertia of the composite object about its central axis in terms of

M_{d}

R

M_{r}

, and

r

. (b) Derive an expression for the fraction of the object’s total kinetic energy that is rotational in terms of the same variables. (c) Let

M_{r}=0.60\,M_{d}

. For what value of

r

(expressed as a multiple of

R

) will the rotational kinetic energy equal the translational kinetic energy? (d) Describe qualitatively how the rotational energy fraction changes as

r

becomes much larger than

R

, and provide a physical explanation for this behavior.