Problem 87 - University Physics with Modern Physics

Question

A metal rod that is 30.0 \(\mathrm{cm}\) long expands by 0.0650 \(\mathrm{cm}\) when its temperature is raised from \(0.0^{\circ} \mathrm{C}\) to \(100.0^{\circ} \mathrm{C}\) . A rod of a different metal and of the same length expands by 0.0350 \(\mathrm{cm}\) for the same rise in temperature. A third rod, also 30.0 \(\mathrm{cm}\) long, is made up of pieces of each of the above metals placed end to end and expands 0.0580 \(\mathrm{cm}\) between \(0.0^{\circ} \mathrm{C}\) and \(100.0^{\circ} \mathrm{C}\) . Find the length of each portion of the composite rod.

Step-by-Step Solution

Answer

The first metal portion is 22.99 cm and the second is 7.01 cm.

Step 1: Understand Thermal Expansion

The thermal expansion of a material can be described by the formula \[\Delta L = \alpha \times L \times \Delta T\]where \( \Delta L \) is the change in length, \( \alpha \) is the coefficient of linear expansion, \( L \) is the original length, and \( \Delta T \) is the change in temperature. Both rods experience the same temperature change from \(0^{\circ} \mathrm{C}\) to \(100^{\circ} \mathrm{C}\), so \( \Delta T = 100^{\circ} \mathrm{C}\).

Step 2: Define Variables for Each Metal

Let the original metal rod with expansion of \(0.0650\, \mathrm{cm}\) have a coefficient of linear expansion \(\alpha_1\), and the second rod with expansion of \(0.0350\, \mathrm{cm}\) have \(\alpha_2\). Each rod has an original length \(L = 30.0\, \mathrm{cm}\).

Step 3: Set Up Equations for Each Metal's Expansion

For the first metal:\[\Delta L_1 = \alpha_1 \times 30.0 = 0.0650\]For the second metal:\[\Delta L_2 = \alpha_2 \times 30.0 = 0.0350\]

Step 4: Solve for Coefficients of Expansion

Solving for \(\alpha_1\) and \(\alpha_2\):\[\alpha_1 = \frac{0.0650}{30.0} = 0.002167\]\[\alpha_2 = \frac{0.0350}{30.0} = 0.001167\]

Step 5: Set Up Equation for Composite Rod

For the composite rod, consisting of lengths \(x\) from the first metal and \(y\) from the second metal:\[x + y = 30.0\, \mathrm{cm}\]\[\alpha_1 \times x + \alpha_2 \times y = 0.0580\]

Step 6: Substitute Coefficients and Solve System of Equations

Substitute the values for \(\alpha_1\) and \(\alpha_2\) and solve:\[0.002167x + 0.001167y = 0.0580\]\[x + y = 30.0\]

Step 7: Solve the First Equation for One Variable

From equation \(x + y = 30.0\), solve for \(y\) in terms of \(x\):\[y = 30.0 - x\]

Step 8: Substitute and Solve for Each Length

Substitute \(y = 30.0 - x\) into the equation:\[0.002167x + 0.001167(30.0 - x) = 0.0580\]Solve for \(x\):\[0.002167x + 0.03501 - 0.001167x = 0.0580\]\[0.001x = 0.0580 - 0.03501\]\[0.001x = 0.02299\]\[x = 22.99\, \mathrm{cm}\]

Step 9: Solve for the Other Length

Using \(y = 30.0 - x\), substitute \(x = 22.99\):\[y = 30.0 - 22.99 = 7.01\, \mathrm{cm}\]

Step 10: Final Solution: Lengths of Each Portion

The length of the portion of the composite rod made of the first metal is approximately \(22.99\, \mathrm{cm}\), and the length made of the second metal is approximately \(7.01\, \mathrm{cm}\).

Key Concepts

Coefficient of Linear Expansion

When it comes to understanding how materials grow larger with heat, the coefficient of linear expansion (\( \alpha \)) plays a crucial role. This coefficient is a material property that describes how much a material's length increases per degree change in temperature. The formula to calculate the change in length, \( \Delta L \), due to thermal expansion is:

• \[ \Delta L = \alpha \times L \times \Delta T \]

Here, \( \alpha \) is the coefficient of linear expansion, \( L \) is the original length, and \( \Delta T \) is the temperature change. For example, in our exercise, the coefficients for two different metals were found to be 0.002167 and 0.001167 respectively, showing how much longer a 1 cm piece of each metal would become when heated by one degree Celsius. These coefficients help engineers and scientists predict and accommodate for expansion in various conditions.

Composite Materials

A composite material is essentially a combination of two or more different materials that are put together to produce a material with different properties than the individual components. In the context of thermal expansion, a composite rod made from two metals can behave uniquely compared to rods made solely from one type of metal. By placing sections of different metals end-to-end, as done in our exercise, the overall expansion behavior depends on both materials' properties.
With composite materials, you can tailor characteristics like thermal expansion to achieve desired performance. For instance, the composite rod in this problem is made of two metals. Each section of the rod expands according to its own coefficient of linear expansion, contributing to the total change in length. This kind of tailored expansion is valuable in situations where control over dimensional changes is important.

Metal Rod Expansion

Metal rods expand when heated due to thermal expansion. The amount they expand depends largely on their coefficient of linear expansion and how much the temperature is increased. In practice, this means a metal rod that was 30 cm at room temperature will lengthen when exposed to heat.
In our exercise, two separate metal rods each had different expansion values when heated by 100°C. The first metal rod expanded by 0.065 cm, while the second expanded by 0.035 cm. This difference is due to their unique physical properties, specifically their coefficients of linear expansion.
Understanding how and why metals expand is critical in designing structures and components that withstand temperature changes. Whether it's compensating for bridges' expansion in the sun or ensuring machinery operates smoothly across temperatures, knowledge of metal expansion is essential.