Question

When the temperature of a copper penny (which is not pure copper) is raised by \(100 \mathrm{C}^{\circ}\), its diameter increases by \(0.18 \%\). Find the percent increase in \((a)\) the area of a face, \((b)\) the thickness, ( \(c\) ) the volume, and \((d)\) the mass of the penny. \((e)\) Calculate its coefficient of linear expansion.

Short answer

The percent increase in: (a) Area of a face is \(0.36%\), (b) Thickness is \(0.18%\), (c) Volume is \(0.54%\), (d) Mass is \(0%\). (e) The coefficient of linear expansion is \(0.0018 / \mathrm{C}^{\circ}\)

Step-by-step solution

Finding the percent increase in the area of a face

The change in area due to expansion is given by \( \Delta A = 2 \alpha A \Delta T \) where \( \Delta A \) is the change in area, \( A \) is the original area, \( \alpha \) is the coefficient of linear expansion, and \( \Delta T \) is the temperature change. Now, we can express \(\alpha\) in terms of the given percent increase in diameter, that is, 0.18%. We'll use that \(2\alpha = 0.18/100\). This gives percent increase in the area as \(2*0.18 = 0.36% \).

Finding the percent increase in the thickness

As the expansion is uniform, the thickness will also increase by the same fraction as diameter. Therefore, the percent increase in the thickness is also \(0.18% \).

Finding the percent increase in the volume

The change in volume due to expansion is given by \( \Delta V = 3\alpha V \Delta T \) where \( V \) is original volume and \( \Delta V \) is change in volume. We can express \( \alpha \) as above. This gives the percent increase in volume as \(3*0.18 = 0.54% \).

Finding the percent increase in mass

The mass of the penny doesn't change with temperature change. So the percent increase in mass is 0%.

Finding the coefficient of linear expansion

The coefficient of linear expansion \( \alpha \) is given by the formula \( \Delta L = \alpha L \Delta T \). Now if we substitute \( \Delta L=d \), \(\alpha=0.18/100\) and \(\Delta T=100 \) we can solve it for \(\alpha\). On solving the equation, we get \( \alpha = 0.0018 / \mathrm{C}^{\circ} \)

Key concepts

Coefficient of Linear Expansion

The coefficient of linear expansion is a quantity that measures how the size of an object changes with a change in temperature. Specifically, it describes how the length of a material changes. This coefficient is usually denoted by the symbol \( \alpha \) and is expressed in reciprocal degrees Celsius \( (1/\mathrm{C}^\circ) \).

Imagine you have a metal rod, and you heat it up; the rod gets longer. How much longer it gets can be predicted by knowing the rod's coefficient of linear expansion and how much the temperature changes. The formula for linear expansion is given by\[ \Delta L = \alpha L \Delta T \], where \( \Delta L \) represents the change in length, \( L \) is the original length, and \( \Delta T \) is the change in temperature. In practical terms, if you know the original length and the temperature change, you can calculate how much the length will change by multiplying these two with the coefficient of linear expansion. Understanding this concept is crucial in fields such as engineering and construction, where materials are subject to changes in temperature.

Percent Increase in Volume

When a solid expands due to an increase in temperature, not only its length changes, but its volume increases as well. The percent increase in volume is a way to quantify how much more space an object occupies as it gets hotter.

For solids, assuming uniform expansion in all directions, the percent increase in volume can be determined by the formula \[ \Delta V = 3\alpha V \Delta T \], where \( \Delta V \) is the change in volume, \( V \) is the original volume, \( \alpha \) is the coefficient of linear expansion, and \( \Delta T \) is the change in temperature. If you multiply the coefficient of linear expansion by 3, and then by the temperature change, you get the percent increase in volume. For instance, if a metal block expands by 0.18% in its length when heated, its volume would increase by approximately 0.54%, as seen in the example problem from the textbook.

Thermal Expansion in Solids

Thermal expansion is a physical property of matter that causes it to change in shape, area, and volume in response to a change in temperature. Solids expand upon heating and contract when cooled because the individual atoms in a solid move more when the temperature goes up, needing more space.

In the context of solids, thermal expansion can result in changes across all dimensions. The linear expansion deals with changes in length, whereas the area and volume expansions are concerned with changes in two-dimensional and three-dimensional space, respectively. It is important to note that the expansion is not necessarily uniform; factors such as material composition, shape, and the manner in which the heat is applied can affect how it expands. For example, in the given problem, the copper penny expands uniformly, which means that its diameter, thickness, and volume all increase by predictable percentages. Engineers and designers must account for thermal expansion to prevent failures in structures, machinery, and other systems that are subjected to temperature fluctuations.