Question

A circular hole in an aluminum plate is$\mathbf{2}\mathbf{.}\mathbf{725}\mathbf{}\mathit{c}\mathit{m}$in diameter at $\mathbf{0}\mathbf{.}\mathbf{000}\mathbf{°}\mathit{C}$.What is its diameter when the temperature of the plate is raised to$\mathbf{100}\mathbf{.}\mathbf{0}\mathbf{°}\mathit{C}$?

Short answer

The diameter of the circular hole when the temperature is raised is$2.731cm$

Step-by-step solution

The given data

• i) Diameter of the circular hole is $D_0=2.725cmor2.725\times {10}^{-2}mat{T}_I=0.0°C$
• ii) Final temperature is$T_f=100.0°C$

Understanding the concept of thermal expansion

Find the change add it to the original diameter, to determine the diameter of a circular hole in an aluminum plate when the temperature in the diameter of the circular hole using the formula for linear expansion. Then of the plate is raised to$100.0°C$.

Formula:

The linear expansion of a body as follows:

$∆L=L\alpha ∆T$ …… (i)

Calculate the value of the diameter

Change in the diameter of the circular hole is due to linear expansion.

From equation (i), write the change in the diameter of aluminum circular hole as:

$∆D=D_0{\alpha }_{Al}∆T$

Here,$$\begin{gathered} ∆T=100-0=100°C \\ {\alpha }_{Al}=23\times {10}^{-6}/{}^{°}C \end{gathered}$$

Hence, the change in diameter is given using the values as:
$$\begin{gathered} ∆D=2.725\times {10}^{-2}\times 23\times {10}^{-6}\times \left(100\right) \\ =62.675\times {10}^{-6}m \\ =62.675\times {10}^{-4}cm \end{gathered}$$

Final diameter is the original diameter plus the change in the diameter. Hence, it is given as:
$$\begin{gathered} D=D_0+∆D \\ =2.725cm+62.675\times {10}^{-4}cm \\ =2.731cm \end{gathered}$$

Hence, the value of the final diameter after temperature raise is 2.731 cm.