Question

The initial length L, change in temperature $\mathbf{△}\mathbf{T}$, and change in length $\mathbf{△}\mathbf{L}$of four rods are given in the following table. Rank the rods according to their coefficients of thermal expansion, greatest first.

Rod L (m) $\mathbf{△}\mathbf{T}\mathbf{}\mathbf{(}\mathbf{C}\mathbf{°}\mathbf{)}$ $\mathbf{△}\mathbf{L}\mathbf{}\mathbf{\left(}\mathbf{m}\mathbf{\right)}$

$$\begin{gathered} \mathbf{a} \\ \mathbf{b} \\ \mathbf{c} \\ \mathbf{d} \end{gathered}$$ $$\begin{gathered} \mathbf{2} \\ \mathbf{1} \\ \mathbf{2} \\ \mathbf{4} \end{gathered}$$ $$\begin{gathered} \mathbf{10} \\ \mathbf{20} \\ \mathbf{10} \\ \mathbf{5} \end{gathered}$$ $$\begin{gathered} \mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{4}} \\ \mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{4}} \\ \mathbf{8}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{4}} \\ \mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{4}} \end{gathered}$$

Short answer

The ranking of the rods according to their coefficients of thermal expansion is${\alpha }_c>{\alpha }_a={\alpha }_b={\alpha }_d$.

Step-by-step solution

The given data

• a) The length of rod a is L= 2m .
• b) The length of rod b is L = 1m .
• c)The length of rod c is L=2m .
• d) The length of rod d is L=4m.
• e)The change in temperature of rod a is$△T=10°C$.
• f) The change in temperature of rod b is $∆T=20°C.$.
• g) The change in temperature of rod c is $∆T=10°C$.
• h) The change in temperature of rod d is $∆T=5°C$
• i) The change in the length of rod a is $∆L=4\times {10}^{-4}m$.
• j) The change in the length of rod b is $∆L=4\times {10}^{-4}m$.
• k) The change in the length of rod c is $∆L=8\times {10}^{-4}m$.

I) The change in the length of rod d is$4\times {10}^{-4}m$

Understanding the concept of thermal expansion

When an object's temperature changes, it expands and grows larger, a process known as thermal expansion. Using Equation 18-9 for the change in the length of the rod, we can find the coefficients of thermal expansion for each rod. Now,

comparing these values for each rod, we can rank the rods according to their coefficient of thermal expansion.

Formula:

The linear expansion of length due to thermal expansion,$∆L=L\alpha ∆T$ …(i)

where,$\alpha$ is the coefficient of linear expansion.

Calculation of the rank according to the coefficient of thermal expansion

From equation (i), thecoefficientof linear expansion is given as:

$\alpha =\frac{∆L}{L∆T}$ …(ii)

Let,${\alpha }_a,{\alpha }_b,{\alpha }_c$and ${\alpha }_d$be the coefficients of linear expansion of rods a, b, c, and d respectively.

Using the given values in equation (ii), we can get the value of coefficient of linear expansion of a as follows:
$$\begin{gathered} {\alpha }_a=\frac{4\times {10}^{4}m}{2m\times {10}^{0}C} \\ =2\times {10}^{-5}{l}^{0}C \end{gathered}$$

Using the given values in equation (ii), we can get the value of coefficient of linear expansion of b as follows:
$$\begin{gathered} {\alpha }_b=\frac{4\times {10}^{-4}m}{1m\times {20}^{0}C} \\ =2\times {10}^{-5}{l}^{0}C \end{gathered}$$

Using the given values in equation (ii), we can get the value of coefficient of linear expansion of c as follows:
$$\begin{gathered} {\alpha }_c=\frac{8\times {10}^{-4}m}{2m\times {10}^{0}C} \\ =4\times {10}^{-5}{l}^{0}C \end{gathered}$$

Using the given values in equation (ii), we can get the value of coefficient of linear expansion of c as follows:
$$\begin{gathered} {\beta }_d=\frac{4\times {10}^{-4}m}{4m\times 5^{0}C} \\ =2\times {10}^{-5}{l}^{0}C \end{gathered}$$

Thus, by comparing the above values we get${\alpha }_c>{\alpha }_a={\alpha }_b={\alpha }_d$

Hence, the ranking of these rods according to their coefficients of expansion is${\alpha }_c>{\alpha }_a={\alpha }_b={\alpha }_d$.