To find the loss in kinetic energy when two spinning discs are brought together, we use the principle of conservation of angular momentum and the formula for kinetic energy. Here's how:

First, because angular momentum before and after they touch must be the same, we have:

$$I_1 \omega_1 + I_2 \omega_2 = (I_1 + I_2) \omega_0$$

where:

• $$I_1$$ and $$I_2$$ are the moments of inertia for the two discs.
• $$\omega_1$$ and $$\omega_2$$ are their angular speeds before contact.
• $$\omega_0$$ is their common angular speed after contact.

Plugging in the given values, we find that:

$$\omega_0 = 8 \mathrm{rad/s}$$

Next, to calculate the loss in kinetic energy, we first find the total kinetic energy before and after they touch:

Before: $$E_1 = \frac{1}{2} I_1 \omega_1^2 + \frac{1}{2} I_2 \omega_2^2 = 216 \,\mathrm{J}$$

After: $$E_2 = \frac{1}{2}(I_1 + I_2) \omega_0^2 = 192 \,\mathrm{J}$$

The loss in kinetic energy ($$\Delta E$$) is the difference:

$$\Delta E = E_1 - E_2 = 24 \,\mathrm{J}$$

So, when the two discs are brought together, the system loses 24 J of kinetic energy.