A bore made in an aluminium block at 34ºC is 3.48cm\(^3\). What is the new bore when the temperature was raised to 340ºC [α\(_a\) = 24 x 10\(^{-6}\)K\(^{-1}...

Answer Details

A bore is a cavity in the aluminium block, and it expands as though it were a solid piece of the same material. Since the bore has a volume (cm³), we use cubical (volume) expansivity, \( \gamma = 3\alpha \).

Given:

• Initial volume, \( V_0 = 3.48 \text{ cm}^3 \)
• Linear expansivity, \( \alpha = 24 \times 10^{-6} \text{ K}^{-1} \)
• Temperature change, \( \Delta T = 340 - 34 = 306 \text{ K} \)

Calculate the cubical expansivity:

\[ \gamma = 3\alpha = 3 \times 24 \times 10^{-6} = 72 \times 10^{-6} \text{ K}^{-1} \]

Apply the volume expansion formula:

\[ V = V_0(1 + \gamma \Delta T) = 3.48(1 + 72 \times 10^{-6} \times 306) \]

\[ V = 3.48(1 + 0.022032) = 3.48 \times 1.022032 \]

\[ V \approx 3.56 \text{ cm}^3 \]

The new bore volume is approximately 3.56 cm³.

Remember: a hole or bore in a material expands exactly as if it were filled with the same material. The linear expansivity given must be converted to cubical expansivity (\( \gamma = 3\alpha \)) whenever the quantity expanding is a volume.