Chapter 13, Problem 89GP

To determine

To Find: The size of the gap.

Answer to Problem 89GP

The size of the gap (difference in radius) is $1.6\times {10}^{-5}\text{ m}$ .

Explanation of Solution

Given info:

Initial temperature, $T_1=20°\text{C}$ .

Final temperature, $T_2=60°\text{C}$ .

Inside diameter of lid, $d=8\text{ cm = }8\times {10}^{-2}\text{ m}$ .

Coefficient of expansion for lid, ${\alpha }_{\text{lid}}=19\times {10}^{-6}\text{/°C}$

Coefficient of expansion for jar, ${\alpha }_{\text{jar}}=9\times {10}^{-6}\text{/°C}$

Initial length, $l_0=2\times \pi \times r$ .

Formula used:

Write the expression for change in the length.

  $\Delta l=l_0\times \alpha \times \Delta T$

Convert the expression for change in length in terms circumference.

  $2\times \pi \times \Delta r=2\times \pi \times r\times \alpha \times \Delta T$

  $\frac{\Delta d}{2}=\frac{d}{2}\times \alpha \times \Delta T$

  $\Delta d=d\times \alpha \times \Delta T$

Here, $l_0$ is the initial length.

  $\alpha$ is the coefficient of linear expansion of the body,

  $\Delta r$ is the change in the radius,

  $\Delta T$ is the change in the temperature.

Calculation:

Calculate the change in diameter for the lid,

  $\begin{array}{l}\Delta d_{\text{lid}}=d_{\text{lid}}\times {\alpha }_{\text{lid}}\times \left(T_2-T_1\right)\\ \Delta d_{\text{lid}}=8\times {10}^{-2}\times 19\times {10}^{-6}\times 40=6.08\times {10}^{-5}\text{ m}\end{array}$

Thus, the change in the radius for the jar.

  $\Delta r_{\text{lid}}=\frac{\Delta d_{\text{lid}}}{2}=3.04\times {10}^{-5}\text{ m}$

Calculate the change in diameter for the jar,

  $\begin{array}{c}\Delta d_{\text{jar}}=d_{\text{jar}}\times {\alpha }_{\text{jar}}\times \left(T_2-T_1\right)\\ \Delta d_{\text{jar}}=8\times {10}^{-2}\times 9\times {10}^{-6}\times 40\\ =2.88\times {10}^{-5}\text{m}\end{array}$

Thus, the change in the radius for the jar,

  $\Delta r_{\text{jar}}=\frac{\Delta d_{\text{jar}}}{2}=1.44\times {10}^{-5}\text{m}$

From these two radii,it can be easily found out the size of gap. Size of gap is also called difference in radius. So, calculate the size of gap as follows.

  $\begin{array}{l}\Delta r=\Delta r_{\text{lid}}-\Delta r_{\text{jar}}\\ \Delta r=\left(3.04\times {10}^{-5}\right)-\left(1.44\times {10}^{-5}\right)\\ \Delta r=1.6\times {10}^{-5}\text{ m}\end{array}$

Conclusion:

Thus, the size of the gap (difference in radius) is $1.6\times {10}^{-5}\text{ m}$ .