Given data

Two coaxial discs have:

• Moment of inertia of first disc: $I_1$
• Moment of inertia of second disc: $I_2=\dfrac{I_1}{2}$

Initial angular velocities:

• First disc: $\omega_1$
• Second disc: $\omega_2=\dfrac{\omega_1}{2}$

After contact, they rotate together with common angular velocity $\omega_f$.

Use conservation of angular momentum

Since no external torque acts on the system, $I_1\omega_1 + I_2\omega_2 = (I_1+I_2)\omega_f$

Substitute the values: $I_1\omega_1 + \frac{I_1}{2}\cdot \frac{\omega_1}{2} = \left(I_1+\frac{I_1}{2}\right)\omega_f$

$I_1\omega_1 + \frac{I_1\omega_1}{4} = \frac{3I_1}{2}\omega_f$

$\frac{5I_1\omega_1}{4} = \frac{3I_1}{2}\omega_f$

Cancelling $I_1$: $\frac{5\omega_1}{4} = \frac{3}{2}\omega_f$

$\omega_f = \frac{5\omega_1}{6}$

Initial rotational kinetic energy

$E_i = \frac{1}{2}I_1\omega_1^2 + \frac{1}{2}I_2\omega_2^2$

$E_i = \frac{1}{2}I_1\omega_1^2 + \frac{1}{2}\cdot \frac{I_1}{2}\cdot \left(\frac{\omega_1}{2}\right)^2$

$E_i = \frac{1}{2}I_1\omega_1^2 + \frac{1}{2}\cdot \frac{I_1}{2}\cdot \frac{\omega_1^2}{4}$

$E_i = \frac{1}{2}I_1\omega_1^2 + \frac{I_1\omega_1^2}{16}$

$E_i = \frac{8+1}{16}I_1\omega_1^2 = \frac{9}{16}I_1\omega_1^2$

Final rotational kinetic energy

$E_f = \frac{1}{2}(I_1+I_2)\omega_f^2$

$E_f = \frac{1}{2}\left(I_1+\frac{I_1}{2}\right)\left(\frac{5\omega_1}{6}\right)^2$

$E_f = \frac{1}{2}\cdot \frac{3I_1}{2} \cdot \frac{25\omega_1^2}{36}$

$E_f = \frac{3I_1}{4}\cdot \frac{25\omega_1^2}{36}$

$E_f = \frac{75}{144}I_1\omega_1^2 = \frac{25}{48}I_1\omega_1^2$

Find $E_f-E_i$

$E_f-E_i = \frac{25}{48}I_1\omega_1^2 - \frac{9}{16}I_1\omega_1^2$

$= \left(\frac{25}{48}-\frac{27}{48}\right)I_1\omega_1^2$

$= -\frac{2}{48}I_1\omega_1^2$

$E_f-E_i = -\frac{I_1\omega_1^2}{24}$

Interpretation

The final energy is less than the initial energy, so energy is lost due to friction during contact.

Hence, $E_f-E_i=-\frac{I_1\omega_1^2}{24}$

Option check

The options list only positive values. The correct algebraic value is negative. If the question intended the loss of energy, then its magnitude is: $\left|E_f-E_i\right|=\frac{I_1\omega_1^2}{24}$ which corresponds to Option A.