Question

A \(20 \mathrm{m}\) steel rod at \(10^{\circ} \mathrm{C}\) is dangling from the edge of a building and is \(2.5 \mathrm{cm}\) from the ground. If the rod is heated to \(110^{\circ} \mathrm{C}\), will the rod touch the ground? (Note: \(a=1.1 \times 10^{-5} \mathrm{K}^{-1}\) ) a. Yes, because it expands by \(3.2 \mathrm{cm}\) b. Yes, because it expands by \(2.6 \mathrm{cm}\) c. No, because it expands by \(2.2 \mathrm{cm}\) d. No, because it expands by \(1.8 \mathrm{cm}\)

Short answer

d. No, because it expands by 2.2 cm

Step-by-step solution

- Understand the relationship of temperature change and expansion

When a material like steel is heated, it expands. The linear expansion \( \Delta L \) of a material can be calculated using the formula \[ \Delta L = L_0 \alpha \Delta T \] where \( L_0 \) is the initial length, \( \alpha \) is the coefficient of linear expansion, and \( \Delta T \) is the change in temperature.

- Calculate the temperature change

The temperature change \( \Delta T \) is the final temperature minus the initial temperature. \[ \Delta T = 110^{\circ} \mathrm{C} - 10^{\circ} \mathrm{C} = 100^{\circ} \mathrm{C} \]

- Apply the expansion formula

Using the formula \[ \Delta L = L_0 \alpha \Delta T \] where \( L_0 = 20 \mathrm{m} = 2000 \mathrm{cm} \), \( \alpha = 1.1 \times 10^{-5} \mathrm{K}^{-1} \), and \( \Delta T = 100 \mathrm{K} \), we get: \[ \Delta L = 2000 \mathrm{cm} \times 1.1 \times 10^{-5} \mathrm{K}^{-1} \times 100 \mathrm{K} = 2.2 \mathrm{cm} \]

- Determine if the rod touches the ground

The rod expands by \( 2.2 \mathrm{cm} \). Since the rod was initially \( 2.5 \mathrm{cm} \) from the ground, and it expands by \( 2.2 \mathrm{cm} \), the new distance from the ground is \( 2.5 \mathrm{cm} - 2.2 \mathrm{cm} = 0.3 \mathrm{cm} \). Therefore, the rod does not touch the ground.

Key concepts

Linear Expansion

When materials like steel are subjected to temperature changes, they expand or contract. This kind of expansion, which happens along the length of the material, is called linear expansion. Essentially, when a material gets hotter, its atoms vibrate more, causing the entire material to increase in size. The increase in length, \( \Delta L \), is directly proportional to the initial length, \( L_0 \), and the change in temperature, \( \Delta T \). The proportional relationship also involves a material-specific factor known as the coefficient of linear expansion, symbolized as \( \ alpha \). Linear expansion is a key concept in thermal physics and is described by the expansion formula.

Temperature Change

Temperature change, denoted as \( \Delta T \), is simply the difference between the final temperature and the initial temperature of the material. In our example of the steel rod, the initial temperature is \( 10^{\circ}\mathrm{C} \) and the final temperature is \( 110^{\circ}\mathrm{C} \). This makes the temperature change:
\[ \Delta T = 110^{\circ}\mathrm{C} - 10^{\circ}\mathrm{C} = 100^{\circ}\mathrm{C} \]
which means that the rod is subjected to an increase of \( 100^{\circ}\mathrm{C} \). Understanding temperature change is crucial because it directly affects how much the material will expand or contract.

Coefficient of Linear Expansion

Materials expand at different rates when heated. The coefficient of linear expansion, \( \ alpha \), helps us quantify this rate. It is a material-specific constant that indicates how much a unit length of the material will expand per degree of temperature change. The unit of \( \ alpha \) is \( \mathrm{\ K^{-1}} \). For example, in the case of our steel rod, the coefficient of linear expansion is \( 1.1 \ times 10^{-5} \mathrm{K^{-1}} \). This tells us that for each degree increase in temperature, each meter of steel will expand by \( 1.1 \ times 10^{-5} \) meters. This factor is essential to accurately calculating the total linear expansion when dealing with temperature changes.

Expansion Formula

The formula used to calculate linear expansion is given by
\( \Delta L = L_0 \ alpha \Delta T \).
In this formula:

\ \
• \( \Delta L \) is the change in length
\ \
• \( L_0 \) is the initial length before expansion
\ \
• \( \alpha \) is the coefficient of linear expansion
\ \
• \( \Delta T \) is the change in temperature

Let's plug in the values from our example:
\[ \Delta L = 2000 \mathrm{cm} \ times 1.1 \ times 10^{-5} \mathrm{K^{-1}} \ times 100 \mathrm{K} = 2.2 \mathrm{cm} \]
Therefore, the steel rod expands by \(2.2 \mathrm{cm} \). Since it was initially \(2.5 \mathrm{cm} \) from the ground, after heating, it will be just \(0.3 \mathrm{cm} \) away from the ground, hence not touching it.