JEE Main 2024 — Rotational Motion Question with Solution

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:

$${\omega }_0=8rad/s$$

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.