JEE MAIN - Physics (2019 - 10th April Morning Slot - No. 12)
Two coaxial discs, having moments of inertia
I1 and I1/2, are rotating with respective angular
velocities $$\omega $$1 and
$$\omega $$1/2
, about their common axis.
They are brought in contact with each other and
thereafter they rotate with a common angular
velocity. If Ef and Ei are the final and initial total
energies, then (Ef - Ei) is:
$${{{I_1}\omega _1^2} \over {24}}$$
$${{{I_1}\omega _1^2} \over {12}}$$
$${3 \over 8}{I_1}\omega _1^2$$
$${{{I_1}\omega _1^2} \over {6}}$$
Explanation
$${E_i} = {1 \over 2}{I_I} \times \omega _1^2 + {1 \over 2}{I \over 2} \times {{\omega _1^2} \over 4}$$
$$ = {{{I_1}\omega _1^2} \over 2}\left( {{9 \over 8}} \right) = {9 \over {16}}{I_1}\omega _1^2$$
$${I_1}{\omega _1} + {{{I_1}{\omega _1}} \over 4} = {{3{I_1}} \over 2}\omega ;{5 \over 4}{I_1}{\omega _1} = {{3{I_1}} \over 2}\omega $$
$$\omega = {5 \over 6}{\omega _1};{E_f} = {1 \over 2} \times {{3{I_1}} \over 2} \times {{25} \over {36}}\omega _1^2$$
$$ = {{25} \over {48}}{I_1}\omega _1^2$$
$$ \Rightarrow {E_f} - {E_i} = {I_1}\omega _1^2{{25} \over {49}} - {{ - 2} \over {48}}{I_2}\omega _1^2$$
$$ = {{25} \over {48}}{I_1}\omega _1^2$$
$$ \Rightarrow {E_f} - {E_i} = {I_1}\omega _1^2\left( {{{25} \over {48}} - {9 \over {16}}} \right) = {{ - 2} \over {48}}{I_1}\omega _1^2$$
$$ = {{ - {I_1}\omega _1^2} \over {24}}$$
$$ = {{{I_1}\omega _1^2} \over 2}\left( {{9 \over 8}} \right) = {9 \over {16}}{I_1}\omega _1^2$$
$${I_1}{\omega _1} + {{{I_1}{\omega _1}} \over 4} = {{3{I_1}} \over 2}\omega ;{5 \over 4}{I_1}{\omega _1} = {{3{I_1}} \over 2}\omega $$
$$\omega = {5 \over 6}{\omega _1};{E_f} = {1 \over 2} \times {{3{I_1}} \over 2} \times {{25} \over {36}}\omega _1^2$$
$$ = {{25} \over {48}}{I_1}\omega _1^2$$
$$ \Rightarrow {E_f} - {E_i} = {I_1}\omega _1^2{{25} \over {49}} - {{ - 2} \over {48}}{I_2}\omega _1^2$$
$$ = {{25} \over {48}}{I_1}\omega _1^2$$
$$ \Rightarrow {E_f} - {E_i} = {I_1}\omega _1^2\left( {{{25} \over {48}} - {9 \over {16}}} \right) = {{ - 2} \over {48}}{I_1}\omega _1^2$$
$$ = {{ - {I_1}\omega _1^2} \over {24}}$$
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