JEE Main 2020 · Physics · Rotational Motion · medium

A uniform disc of mass 2 kg and radius 0.5 m rotates freely about its fixed vertical axis at 4 rad s^-1. A beetle of mass 0.5 kg sits on its rim and…

Single correct option

A uniform disc of mass

2\ \mathrm{kg}

and radius

0.5\ \mathrm{m}

rotates freely about its fixed vertical axis at

4\ \mathrm{rad\,s^{-1}}

. A beetle of mass

0.5\ \mathrm{kg}

sits on its rim and slowly crawls to the centre. The work done by the beetle (the increase in the kinetic energy of the system) is

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Answer: (C) $1.5\ \mathrm{J}$

No external torque acts about the axis, so angular momentum is conserved. Initially

I_1 = \tfrac12MR^2 + mR^2 = 0.25 + 0.125 = 0.375\ \mathrm{kg\,m^2}

; finally

I_2 = \tfrac12MR^2 = 0.25\ \mathrm{kg\,m^2}

. So

\omega_2 = I_1\omega_1/I_2 = 6\ \mathrm{rad\,s^{-1}}

. Kinetic energies:

K_1 = \tfrac12(0.375)(16) = 3\ \mathrm{J}

and

K_2 = \tfrac12(0.25)(36) = 4.5\ \mathrm{J}

. The extra

1.5\ \mathrm{J}

is supplied by the beetle's internal (muscular) work, although angular momentum is conserved.

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