Linear Thermal Expansion: Videos & Practice Problems

Linear thermal expansion describes how materials expand in size with temperature increases. For one-dimensional objects like rods, the change in length (ΔL) is directly proportional to the change in temperature (ΔT) and the initial length (L₀). The equation is expressed as $\mathrm{\Delta L}=\alpha L₀\mathrm{\Delta T}$, where α is the linear expansion coefficient. Understanding this concept is crucial for applications like construction, where materials must accommodate thermal changes without damage.

Linear Thermal Expansion

Linear thermal expansion refers to the phenomenon where materials expand in size when their temperature increases. This is particularly noticeable in one-dimensional objects, such as metal rods. When the temperature of a rod changes, its length also changes, which can be quantified using the relationship between temperature change and length change.

The fundamental equation governing linear thermal expansion is given by:

$$\Delta L = \alpha \cdot L_0 \cdot \Delta T$$

In this equation, ΔL represents the change in length, α is the linear expansion coefficient (a measure of how easily a material expands), L₀ is the initial length of the rod, and ΔT is the change in temperature. The linear expansion coefficient is typically expressed in units of 1/K or 1/°C.

For example, consider an aluminum rod with an initial length of 50 meters at a temperature of 20°C. If the temperature increases to 35°C, the change in temperature (ΔT) is 15°C. Given that the linear expansion coefficient for aluminum is approximately 2.4 × 10^-5 1/°C, we can calculate the change in length:

$$\Delta L = (2.4 \times 10^{-5}) \cdot 50 \cdot 15 = 0.018 \text{ meters}$$

This result indicates that the rod lengthens by about 1.8 centimeters, which is a small but measurable change.

To find the final length of the rod after heating it to a higher temperature, such as 50°C, we can use the rearranged equation:

$$L_f = L_0 \cdot (1 + \alpha \cdot \Delta T)$$

In this case, the change in temperature from 20°C to 50°C is 30°C. Plugging in the values:

$$L_f = 50 \cdot (1 + (2.4 \times 10^{-5}) \cdot 30) = 50.036 \text{ meters}$$

This calculation shows that the final length of the rod is approximately 50.036 meters after the temperature increase.

Understanding linear thermal expansion is crucial in various applications, such as construction and engineering, where materials must accommodate temperature changes without causing structural damage.

On a very cold day at a temperature of –12°C, a power line made of aluminum between two support towers measures exactly 150.56m. You go out on a hot day and measure the power line to be exactly 150.71m. What is the temperature (in °C) outside? The linear expansion coefficient of aluminum is 2.4×10^-5.

• A. 29.5 °C
• B. –11.8°C
• C. 41.7 °C

Answer: (A)

Expanding Steel Measuring Tape

In this problem, we explore the concept of thermal expansion, particularly how it affects the accuracy of measurements taken with a steel measuring tape calibrated at a specific temperature. The key point is that the tape is accurate at 20 degrees Celsius, meaning that when it shows a measurement of 50.000 meters, that is indeed the true length. However, as the temperature increases to 40 degrees Celsius, the steel tape expands due to thermal expansion, which alters its physical length while the markings remain unchanged.

To understand this, we can visualize the situation: at 20 degrees Celsius, the tape measures 50.000 meters accurately. When the temperature rises to 40 degrees Celsius, the actual length of the tape increases slightly, but the ruler still indicates 50.000 meters. This discrepancy arises because the physical properties of the steel change with temperature, leading to an expansion that affects the measurement.

To calculate the actual length at the higher temperature, we use the formula for linear expansion:

\( L_f = L_0 (1 + \alpha \Delta T) \)

In this equation, \( L_f \) represents the actual length at the new temperature, \( L_0 \) is the original length (50.000 meters), \( \alpha \) is the linear expansion coefficient for steel (approximately \( 1.2 \times 10^{-5} \, \text{°C}^{-1} \)), and \( \Delta T \) is the change in temperature.

Here, the change in temperature \( \Delta T \) is calculated as follows:

\( \Delta T = 40 \, \text{°C} - 20 \, \text{°C} = 20 \, \text{°C} \)

Substituting these values into the formula gives:

\( L_f = 50.000 \, \text{m} \times (1 + (1.2 \times 10^{-5}) \times 20) \)

Calculating this results in:

\( L_f = 50.000 \, \text{m} \times (1 + 0.00024) = 50.012 \, \text{m} \)

This means that at 40 degrees Celsius, while the ruler still reads 50.000 meters, the actual distance measured is 50.012 meters. This illustrates the importance of considering temperature effects on measurement accuracy, especially in precision tasks where thermal expansion can lead to significant discrepancies.

Here’s what students ask on this topic:

What is linear thermal expansion and how is it calculated?

Linear thermal expansion refers to the increase in length of a material when its temperature is increased. The change in length (ΔL) is directly proportional to the change in temperature (ΔT) and the initial length (L₀). The formula to calculate this is:

$\mathrm{\Delta L}=\alpha L_0\mathrm{\Delta T}$

where α is the linear expansion coefficient, which varies for different materials. This coefficient is typically given in units of 1/K or 1/°C. Understanding this concept is crucial for applications like construction, where materials must accommodate thermal changes without damage.

What is the linear expansion coefficient and how is it used?

The linear expansion coefficient (α) is a material-specific constant that quantifies how much a material expands per degree change in temperature. It is used in the formula for linear thermal expansion:

$\mathrm{\Delta L}=\alpha L_0\mathrm{\Delta T}$

where ΔL is the change in length, L₀ is the initial length, and ΔT is the change in temperature. The units of α are typically 1/K or 1/°C. This coefficient helps predict how much a material will expand or contract with temperature changes, which is essential for designing structures that can withstand thermal stress.

How do you calculate the final length of a rod after thermal expansion?

To calculate the final length (L_f) of a rod after thermal expansion, you can use the formula:

$L_f=L_0(1+\alpha \mathrm{\Delta T})$

where L₀ is the initial length, α is the linear expansion coefficient, and ΔT is the change in temperature. This formula accounts for the initial length and the additional length gained due to thermal expansion. It is useful in engineering and construction to ensure that materials can accommodate temperature changes without causing structural issues.

Why is it important to consider linear thermal expansion in construction?

Considering linear thermal expansion in construction is crucial because materials expand and contract with temperature changes. If this expansion is not accounted for, it can lead to structural damage, such as cracks in concrete, warping of metal beams, or failure of joints. For example, gaps are often left between sections of sidewalks and bridges to allow for expansion on hot days. By understanding and planning for thermal expansion, engineers can design structures that remain safe and functional under varying temperature conditions.

Can you use Celsius and Kelvin interchangeably in thermal expansion calculations?

Yes, you can use Celsius and Kelvin interchangeably in thermal expansion calculations when dealing with temperature changes (ΔT). This is because the size of one degree Celsius is equal to one Kelvin. Therefore, the change in temperature (ΔT) will be the same whether measured in Celsius or Kelvin. However, this interchangeability only applies to temperature differences, not absolute temperatures.