Question

The Pyrex glass mirror in a telescope has a diameter of 170 in.The temperature ranges from$\mathbf{-}\mathbf{16}\mathbf{°}\mathbf{\text{C}}$to$\mathbf{32}\mathbf{°}\mathbf{\text{C}}$on the location of the telescope. What is the maximum change in the diameter of the mirror, assuming that the glass can freely expand and contract?

Short answer

The maximum change in the diameter of a mirror is$6.63\times {10}^{-4}\text{m}$

Step-by-step solution

Identification of given data

1. The initial length of diameter of Pyrex glass mirror is$L_0=170\text{inor}4.318\text{m}$
2. The temperature changes from$-16°\text{Cto}32°\text{C}.$

Understanding the concept of heat transfer 

The length, width, height, or volume of the body vary when the temperature has an impact on it. We use the concept of linear thermal expansion to solve this problem. We are assuming that glass can freely expand and contract. So, the glass will expand uniformly as the temperature increases.

Formula:

The linear expansion of the body due to thermal radiation,$\Delta L=L_0\times \alpha \times \Delta t$ …(i)

Determining the change in diameter

The temperature difference from the given data can be given as:

$$\begin{gathered} \Delta t={\left(32-(-16)\right)}^{\circ }C \\ ={48}^0\text{C} \end{gathered}$$

Linear coefficient of thermal expansion for pyrex is$\alpha =3.2\times {10}^{-6}{/}^0\text{C}$

Using equation (i) and given values, the maximum change in the diameter of the mirror is given as:

$\begin{array}{c}\Delta L=4.318\text{\hspace{0.17em}m}\times 3.2\times {10}^{-6}{/}^0\text{C}\times {48}^0\text{C}\\ =6.63\times {10}^{-4}\text{m}\end{array}$

Hence, the value of the change in diameter is$6.63\times {10}^{-4}\text{m}$