Question

Two coaxial discs having moments of inertia ${I_1}$ and $\dfrac{{{I_1}}}{2}$ are rotating with respective angular velocities ${\omega _1}$ and $\dfrac{{{\omega _1}}}{2}$ , about their common axis. They are brought in contact with each other and thereafter they rotate with a common angular velocity. If ${E_f}$ and ${E_i}$ are the final and initial total energies, then $\left( {{E_f} - {E_i}} \right)$ is:
\left( a \right)$$$\dfrac{{{I_1}\omega _1^2}}{{12}}$$ \left( b \right)$$\dfrac{3}{8}{I_1}\omega _1^2$$\left( c \right)$$\dfrac{{{I_1}\omega _1^2}}{6}$$\left( d \right)$$\dfrac{{ - {I_1}\omega _1^2}}{{24}}$

Answer 1

Hint So in this question, it is given that there are the two-discs and they are having a moment of inertia and having some angular velocities. Now they are saying that it will be brought in contact with one another and then it gets rotated. By using the equation given on conservation of angular momentum. And so by doing this we get the answer.
Formula:
Conservation of angular momentum,
${L_i} = {L_f}$
And Kinetic energy,
$\Rightarrow \dfrac{1}{2}I{\omega ^2}$
Where,
$I$ , will be the inertia and $\omega$ will be the angular velocity.

Complete Step By Step Solution As from the question it is clear that we have to find the difference between the two energies which is final and the initial.
As we know energy in terms of inertia for both the disc will be equal to
$\Rightarrow E = \dfrac{1}{2}{I_1} \times \omega _1^2 + \dfrac{1}{2}\dfrac{{{I_2}}}{2} \times \dfrac{{\Omega _1^2}}{4}$
Now putting the values from the question,
$\Rightarrow \dfrac{{{I_1}\omega _1^2}}{2}\left( {\dfrac{9}{8}} \right) = \dfrac{9}{{16}}{I_1}\omega _1^2 = {E_I}$
Now on further solving the above equation,
We get
$\Rightarrow {I_1}{\omega _1} = \dfrac{{{I_1}\omega _1^{}}}{4} = \dfrac{{3{I_1}}}{2}\omega$
And from here
$\Rightarrow \omega = \dfrac{5}{6}{\omega _1}$
Now we will calculate the final energy, so for this final energy will be equal to
$\Rightarrow {E_f} = \dfrac{1}{2} \times \dfrac{{3{I_1}}}{2} \times \dfrac{{25}}{{36}}\omega _1^2$
After solving the above equation, we will get the equation as
$\Rightarrow \dfrac{{25}}{{48}}{I_1}\omega _1^2$
Now we will calculate the difference between that will be represented as $\left( {{E_f} - {E_i}} \right)$ ,
So equating the value we obtained from the above calculation, we get
$\Rightarrow {E_f} - {E_i} = {I_1}\omega _1^2\left( {\dfrac{{25}}{{48}} - \dfrac{9}{{16}}} \right)$
Again on further solving the above equation, we get
$\Rightarrow \dfrac{{ - 2}}{{48}}{I_1}\omega _1^2$
And after simplifying, we get
$\Rightarrow \dfrac{{ - {I_1}\omega _1^2}}{{24}}$
Therefore, $\dfrac{{ - {I_1}\omega _1^2}}{{24}}$ , is the difference between the final and initial energies. Hence the option $D$ is the correct one among all the four options given here.

Note The Law of Conservation of Angular Momentum is extremely important: it explains how explosions work, why our planet still spins, why we orbit the sun, and why, when two things crash into each other, they sometimes bounce.