Question

An aluminum-alloy rod has a length of 10.00 cmat 20.00⁰Cand a length of 10.015 cmat the boiling point of water.

(a) What is the length of the rod at the freezing point of water?

(b) What is the temperature if the length of the rod is 10.009 cm?

Short answer

1. The length of the rod at the freezing point of water is 9.996 cm
2. The temperature value is${68}^{\mathrm{o}}\mathrm{C}$

Step-by-step solution

The given data

1. Aluminum-alloy rod length is ${\mathrm{L}}_1=10.00\mathrm{cm}$at${\mathrm{T}}_1={20}^{\mathrm{o}}\mathrm{C}$
2. Aluminum-alloy rod length is${\mathrm{L}}_2=10.015\mathrm{cm}$ at${\mathrm{T}}_2={100}^{\mathrm{o}}\mathrm{C}$

Understanding the concept of linear expansion

Thermal expansion can occur due to an increase in temperature. It has three types’ i.e. linear expansion, surface expansion, and volume expansion. For the given problem, we have to use the formula for linear expansion to find the length of the rod and the temperature of the rod.

Formula:

The linear expansion of a body, $∆\mathrm{L}=\mathrm{L\alpha }∆\mathrm{T}$ …(i)

Step 3:(a) Calculation of length of the rod at the freezing point

Using equation (i), the coefficient of linear expansion for aluminum-alloy is given as:

$\begin{array}{rcl}\mathrm{\alpha }& =& \frac{∆\mathrm{L}}{\mathrm{L}∆\mathrm{T}}\\ & =& \frac{10.015\mathrm{cm}-10.00\mathrm{cm}}{10.0\mathrm{cm}\times {\left(100-20\right)}^{\mathrm{o}}\mathrm{C}}\\ & =& \frac{0.015\mathrm{cm}}{10.0\mathrm{cm}\times {80}^{\mathrm{o}}\mathrm{C}}\\ & =& 1.88\times {10}^{-5}/{}^{\mathrm{o}}\mathrm{C}\end{array}$

We have to calculate the change in length of the rod at freezing point of the water so that the change in temperature is given as:

$\begin{array}{rcl}∆\mathrm{T}& =& 0^{\mathrm{o}}\mathrm{C}-{100}^{\mathrm{o}}\mathrm{C}\\ & =& -{100}^{\mathrm{o}}\mathrm{C}\end{array}$

Length of rod at${100}^{\mathrm{o}}\mathrm{C}$is${\mathrm{L}}_2=10.015\mathrm{cm}$

So, the change in linear expansion of the rod is given as:

$\begin{array}{rcl}∆\mathrm{L}& =& 10.015\mathrm{cm}\times 1.88\times {10}^{-5}/{}^{\mathrm{o}}\mathrm{C}\times \left(-{100}^{\mathrm{o}}\mathrm{C}\right)\\ & =& -1.88\times {10}^{-2}\mathrm{cm}\end{array}$

Total length at 0⁰C is given as the sum of original length and the expanded length, that is:

$\begin{array}{rcl}{\mathrm{L}}^{\text{'}}& =& \mathrm{L}+∆\mathrm{L}\\ & =& 10.015\mathrm{cm}-1.88\times {10}^{-2}\mathrm{cm}\\ & =& 10.015\mathrm{cm}-0.0188\mathrm{cm}\\ & =& 9.996\mathrm{cm}\end{array}$

Hence, the value of the length of the rod is 9.996 cm

(b) Calculation of the temperature

The length of the rod is given as$\mathrm{L}=10.009\mathrm{cm}$

But we know that$∆\mathrm{L}=10.009\mathrm{cm}-10.00=0.009\mathrm{cm}$

From equation (i), the change in the temperature due to expansion is given as:

$\begin{array}{rcl}0.009\mathrm{cm}& =& 10.00\mathrm{cm}\times 1.88\times {10}^{-5}/{}^{\mathrm{o}}\mathrm{C}\times ∆\mathrm{T}\\ ∆\mathrm{T}& =& \frac{0.009\mathrm{cm}}{10.00\mathrm{cm}\times 1.88\times {10}^{-5}/{}^{\mathrm{o}}\mathrm{C}}\\ & =& 47.{87}^{\mathrm{o}}\mathrm{C}\\ & \approx & {48}^{\mathrm{o}}\mathrm{C}\\ & =& \end{array}$

But,$∆\mathrm{T}={\mathrm{T}}_{\mathrm{x}}-{20}^{\mathrm{o}}\mathrm{C}$,

where, T_x is the temperature at which the length of the rod is$\mathrm{L}=10.009\mathrm{cm}$

$\begin{array}{rcl}{\mathrm{T}}_{\mathrm{x}}-{20}^{\mathrm{o}}\mathrm{C}& =& {48}^{\mathrm{o}}\mathrm{C}\\ {\mathrm{T}}_{\mathrm{x}}& =& {48}^{\mathrm{o}}\mathrm{C}+{20}^{\mathrm{o}}\mathrm{C}\\ {\mathrm{T}}_{\mathrm{x}}& =& {68}^{\mathrm{o}}\mathrm{C}\end{array}$

Hence, the value of the required temperature is${68}^{\mathrm{o}}\mathrm{C}$.