Conceptual Example 5 provides background for this problem. A hole is drilled through a copper plate whose temperature is 11 °C.

(a) When the temperature of the plate is increased, will the radius of the hole be larger or smaller than the radius at 11 °C? Why?

(b) When the plate is heated to 110 °C, by what fraction Δr/r0 will the radius of the hole change?

Conceptual Example 5

Interactive Figure 12.17a shows eight square tiles that are attached together and arranged to form a square pattern with a hole in the center. If the tiles are heated, does the size of the hole (a) decrease or (b) increase?

Reasoning

We can analyze this problem by disassembling the pattern into separate tiles, heating them, and then reassembling the pattern. What happens to each of the individual tiles can be explained using what we know about linear expansion.

Answer (a) is incorrect. When a tile is heated both its length and width expand. It is tempting to think, therefore, that the hole in the pattern decreases as the surrounding tiles expand into it. However, this is not correct, because any one tile is prevented from expanding into the hole by the expansion of the tiles next to it.

Answer (b) is correct. Since each tile expands upon heating, the pattern also expands, and the hole along with it, as shown in Interactive Figure 12.17b. In fact, if we had a ninth tile that was identical to the others and heated it to the same extent, it would fi t exactly into the hole, as Interactive Figure 12.17c indicates. Thus, not only does the hole expand, it does so exactly as each of the tiles does. Since the ninth tile is made of the same material as the others, we see that the hole expands just as if it were made of the material of the surrounding tiles.

The thermal expansion of the hole and the surrounding material is analogous to a photographic enlargement: everything is enlarged, including holes.

Transcribed Image Text:

Expanded hole Hole 9th tile (heated) (a) Unheated (b) Heated (c)