Question

A medical device used for handling tissue samples has two metal screws, one \(20.0 \mathrm{~cm}\) long and made from brass \(\left(\alpha_{\mathrm{b}}=18.9 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\right)\) and the other \(30.0 \mathrm{~cm}\) long and made from aluminum \(\left(\alpha_{\mathrm{a}}=23.0 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\right)\). A gap of \(1.00 \mathrm{~mm}\) exists between the ends of the screws at \(22.0^{\circ} \mathrm{C}\). At what temperature will the two screws touch?

Short answer

The linear expansion coefficients for brass and aluminum are 18.9 x 10^-6 ºC^-1 and 23.0 x 10^-6 ºC^-1 respectively. Answer: The two screws will touch when the temperature reaches approximately 22.02ºC.

Step-by-step solution

Identify the variables given

In this exercise, we are given: - The initial length of brass screw (\(L_{\mathrm{b}}=20.0 \mathrm{~cm}\)). - The initial length of aluminum screw (\(L_{\mathrm{a}}=30.0 \mathrm{~cm}\)). - The linear expansion coefficient for brass (\(\alpha_{\mathrm{b}}=18.9 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\)). - The linear expansion coefficient for aluminum (\(\alpha_{\mathrm{a}}=23.0 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\)). - The gap between the ends of the screws at \(22.0^{\circ} \mathrm{C}\) (\(d=1.00 \mathrm{~mm}\)). We need to find the temperature at which the two screws will touch.

Convert all measurements to meters

To make calculations easier, convert all measurements to meters: - Brass screw length: \(L_{\mathrm{b}}=0.20 \mathrm{~m}\) - Aluminum screw length: \(L_{\mathrm{a}}=0.30 \mathrm{~m}\) - Gap: \(d=0.001 \mathrm{~m}\)

Calculate the expansion of each screw

The expansion of an object due to temperature change can be calculated using the formula: $$\Delta L = L \cdot \alpha \cdot \Delta T$$ Where \(\Delta L\) is the change in length, \(L\) is the initial length, \(\alpha\) is the linear expansion coefficient, and \(\Delta T\) is the temperature change. For the brass screw, we have: $$\Delta L_{\mathrm{b}} = L_{\mathrm{b}} \cdot \alpha_{\mathrm{b}} \cdot \Delta T$$ For the aluminum screw, we have: $$\Delta L_{\mathrm{a}} = L_{\mathrm{a}} \cdot \alpha_{\mathrm{a}} \cdot \Delta T$$

Set up an equation for when the screws will touch

When the screws touch, the sum of their expansions will be equal to the initial gap between them: $$\Delta L_{\mathrm{b}} + \Delta L_{\mathrm{a}} = d$$ Substituting the expansion formulas from step 3, we get: $$L_{\mathrm{b}} \cdot \alpha_{\mathrm{b}} \cdot \Delta T + L_{\mathrm{a}} \cdot \alpha_{\mathrm{a}} \cdot \Delta T = d$$

Solve for the temperature change \(\Delta T\)

Factor out \(\Delta T\) from the equation in step 4: $$\Delta T (L_{\mathrm{b}} \cdot \alpha_{\mathrm{b}} + L_{\mathrm{a}} \cdot \alpha_{\mathrm{a}}) = d$$ Now, divide both sides by \((L_{\mathrm{b}} \cdot \alpha_{\mathrm{b}} + L_{\mathrm{a}} \cdot \alpha_{\mathrm{a}})\) to solve for \(\Delta T\), using the values in meters from step 2: $$\Delta T = \frac{d}{L_{\mathrm{b}} \cdot \alpha_{\mathrm{b}} + L_{\mathrm{a}} \cdot \alpha_{\mathrm{a}}} = \frac{0.001}{(0.20 \cdot 18.9 \cdot 10^{-6}) + (0.30 \cdot 23.0 \cdot 10^{-6})} \approx 0.019668 \mathrm{~K}$$

Calculate the final temperature

Now that we have the temperature change, we can add it to the initial temperature of \(22.0^{\circ}\)C to find the final temperature: $$T_{\mathrm{final}} = T_{\mathrm{initial}} + \Delta T = 22.0 + 0.019668 = 22.01967^{\circ} \mathrm{C}$$ The two screws will touch when the temperature reaches approximately \(22.02^{\circ} \mathrm{C}\).

Key concepts

Linear Expansion

When most materials are heated, they expand in size. This behavior is predictable and is referred to as linear expansion. Linear expansion describes how the length of an object changes with temperature. It is important in fields that rely heavily on precise measurements, such as engineering and materials science.

Linear expansion occurs in one dimension and affects only the length of the material. Imagine a long thin metal rod expanding as the surrounding temperature rises. This is the simplest model of linear expansion.

• The amount by which the length changes depends on the original length of the material.
• It also depends on the degree of temperature change.
• Different materials will expand by different amounts, even for the same temperature change.

By understanding the principles of linear expansion, engineers can design bridges, tools, and devices to ensure they work correctly over a range of temperatures.

Coefficient of Linear Expansion

The coefficient of linear expansion (\(\alpha\)) is a numerical value that represents how much a material expands per degree of temperature change. Every material has a unique coefficient, which is usually expressed in \((^{\circ}C)^{-1}\). Knowing the coefficient allows us to predict how much an object will expand or contract when its temperature changes.

This coefficient provides critical information when planning for materials to be used in environments that undergo temperature fluctuations. Here's how it is applied:

• A higher coefficient means the material expands more with temperature increase.
• For example, aluminum has a higher coefficient than brass, meaning it will expand more when heated.
• This knowledge helps in selecting materials for specific applications where thermal expansion might be critical, such as in precision instruments or structural components.

With the coefficient of linear expansion, we can calculate the change in length of a material using formulas that account for this property.

Temperature Change Calculation

When calculating how much a material will expand or contract due to temperature changes, it's essential to compute the necessary temperature change (\(\Delta T = T_{\text{final}} - T_{\text{initial}}\)). This involves understanding the relationship between material length, temperature, and expansion coefficients.

The formula \(\Delta L = L \cdot \alpha \cdot \Delta T\) is used, where:

• \(\Delta L\) is the change in length of the material.
• \(L\) is the initial length.
• \(\alpha\) is the coefficient of linear expansion.

To find the temperature at which two objects will expand to meet each other, like the metal screws in the provided problem, we calculate the necessary temperature change. This involves setting up an equation that considers the sum of the expansions being equal to the initial gap:

• The gap closure temperature can be found using \(\Delta T = \frac{d}{(L_{\text{b}} \cdot \alpha_{\text{b}} + L_{\text{a}} \cdot \alpha_{\text{a}})}\).
• The above formula helps derive the specific temperature at which the objects will touch by balancing the expansion of different materials.

This calculation keeps in mind the initial separation and utilizes the coefficients specific to each material, ensuring accurate predictions for real-world applications.