Question

A U.S. penny has a diameter of 1.9000 cm at 20.0ºC. The coin is made of a metal alloy (mostly zinc) for which the coefficient of linear expansion is$\mathbf{2}\mathbf{.}\mathbf{6}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}{\mathbf{K}}^{\mathbf{-}\mathbf{1}}$ . What would its diameter be on a hot day in Death Valley (48.0°C)? On a cold night in the mountains of Greenland (-53ºC)?

Short answer

The diameter value on a hot day in Death Valley is $1.9014\mathrm{cm}$and the diameter value on a cold night in the Greenland is$1.8964\mathrm{cm}$.

Step-by-step solution

Concept of Linear Expansion in solids.

The linear change in length when the temperature change is given by the linear expansion expression,

$\mathbf{L}\mathbf{=}{\mathbf{L}}_{\mathbf{0}}\left(1+\alpha ∆T\right)$ ...(i)

Here, L is the expanded length, L₀ is the original length, α is the coefficient of expansion of the material and T is the absolute temperature.

 Determination of the diameter value on a hot day in Death Valley is and the diameter value on a cold night in the Greenland.

For Death Valley:

Change in temperature is,

$$\begin{gathered} ∆T=\left(48.0°C-20.0°C\right) \\ =28.0°C \end{gathered}$$

Use equation, $∆D=D_0\alpha ∆T$to find the changed diameter after the temperature increases.

$$\begin{gathered} ∆D=\left(2.6\times {10}^{-5}{\left(\mathrm{C}°\right)}^{-1}\right)\left(1.90\mathrm{cm}\right)\left(28.0\mathrm{C}°\right) \\ =1.4\times {10}^{-3}\mathrm{cm} \end{gathered}$$

Thus, the changed length is,

$$\begin{gathered} D=D_0+∆D \\ =\left(1.9000+1.4\times {10}^{-3}\right)cm \\ =1.9014\mathrm{cm} \end{gathered}$$

For Greenland,

Change in temperature is,

$$\begin{gathered} ∆T=\left(-53.0°C-20.0°C\right) \\ =-73°C \end{gathered}$$

Use equation, $∆D=D_0\alpha ∆T$to find the changed diameter after the temperature increases.

$$\begin{gathered} ∆D=\left(2.6\times {10}^{-5}{\left(C°\right)}^{-1}\right)\left(1.90cm\right)\left(-73.0C°\right) \\ =-3.6\times {10}^{-3}cm \end{gathered}$$

Thus, the changed length is,

$$\begin{gathered} D=D_0+∆D \\ =\left(1.9000-3.6\times {10}^{-3}\right)cm \\ =1.8964\mathrm{cm} \end{gathered}$$