Center of Mass and Collision — JEE Main practice

Given below are two statements :

Statement I : For a mechanical system of many particles total kinetic energy is the sum of kinetic energies of all the particles.

Statement II : The total kinetic energy can be the sum of kinetic energy of the center of mass w.r.t to the origin and the kinetic energy of all the particles w.r.t. the center of mass as the reference.

In the light of the above statements, choose the correct answer from the options given below :

A small bob $A$ of mass $m$ is attached to a massless rigid rod of length $1\,\mathrm{m}$ pivoted at point $P$ and kept at an angle of $60^{\circ}$ with vertical as shown in figure. At distance of $1\,\mathrm{m}$ below point $P$, an identical bob $B$ is kept at rest on a smooth horizontal surface that extends to a circular track of radius $R$ as shown in figure. If bob $B$ just manages to complete the circular path of radius $R$ upto a point $Q$ after being hit elastically by bob $A$, then radius $R$ is ____ m.

In a perfectly inelastic collision, two spheres made of the same material with masses $15\,\mathrm{kg}$ and $25\,\mathrm{kg}$, moving in opposite directions with speeds of $10\,\mathrm{m/s}$ and $30\,\mathrm{m/s}$, respectively, strike each other and stick together. The rise in temperature (in $^{\circ}\mathrm{C}$), if all the heat produced during the collision is retained by these spheres, is :

(specific heat of sphere material $31\,\mathrm{cal/kg\cdot^{\circ}C}$ and $1\,\mathrm{cal}=4.2\,\mathrm{J}$)