When a metallic disc with a circular hole at the centre is heated, the material of the disc undergoes thermal expansion. Thermal expansion is the tendency of matter to change its shape, area, and volume in response to a change in temperature.

Thermal Expansion of Materials

Most materials expand when heated and contract when cooled. This happens because the atoms in the material vibrate more vigorously at higher temperatures, pushing each other further apart on average. This leads to an increase in the overall size of the material.

For a solid, this expansion occurs in all dimensions – length, width, and thickness (or radius and circumference in the case of a disc).

Effect of Heating on the Circular Disc and Hole

Consider the circular metallic disc. When it is heated, every part of the disc expands. The material that forms the disc expands outwards from the center.

Now, let's think about the hole in the centre. The hole is essentially an absence of material within the disc. However, the behavior of the hole can be understood by imagining that the hole is filled with the same material as the disc, and this entire larger disc (with no hole) is heated. As the larger disc expands uniformly, the part corresponding to the hole would also expand.

Therefore, when the disc with a hole is heated, the material around the hole expands away from the center of the hole. This causes the circumference of the hole to increase, and consequently, the diameter of the hole also increases.

An analogy often used is that of a photograph being uniformly enlarged. Everything in the photograph, including any empty spaces or holes within the image, gets larger proportionally. Similarly, when a metallic disc with a hole is heated, it expands as if it were a photograph being enlarged – every dimension scales up.

Let the original diameter of the hole be $D_0$ at temperature $T_0$. When the temperature increases to $T$, the new diameter $D$ will be given by the formula for linear expansion, applied to the diameter of the hole:

$\Delta D = \alpha D_0 \Delta T$

$D - D_0 = \alpha D_0 (T - T_0)$

$D = D_0 (1 + \alpha \Delta T)$

where $\alpha$ is the coefficient of linear expansion of the metal, and $\Delta T = T - T_0$ is the change in temperature. Since $\alpha$ is positive for most materials and $\Delta T$ is positive when heating, $\Delta D$ is positive, meaning the diameter increases.

Conclusion

Heating a circular metallic disc causes thermal expansion of the material. This expansion is isotropic (same in all directions) for homogeneous materials like metals. The hole in the centre behaves as if it were filled with the same material expanding along with the rest of the disc. Thus, the diameter of the hole increases upon heating.

The final answer is $\boxed{Increase}$.