Question

A steel track measures \(25 \mathrm{~m}\) at \(20^{\circ} \mathrm{C}\). At \(70{ }^{\circ} \mathrm{C}\) it measures \(25.01 \mathrm{~m}\). Calculate the percentage change in the length of the track as its temperature changes from \(20^{\circ} \mathrm{C}\) to \(70{ }^{\circ} \mathrm{C}\).

Short answer

Answer: 0.04%

Step-by-step solution

Calculate the change in length

To find the change in length, we need to subtract the initial length from the final length: Change in length = Final length - Initial length Change in length = 25.01 m - 25 m Change in length = 0.01 m

Calculate the percentage change in length

To find the percentage change in length, we will divide the change in length by the initial length and multiply by 100: Percentage change = (Change in length / Initial length) * 100 Percentage change = (0.01 m / 25 m) * 100 Percentage change = 0.0004 * 100 Percentage change = 0.04 % So, the percentage change in the length of the steel track as its temperature changes from 20°C to 70°C is 0.04%.

Key concepts

Percentage Change

When dealing with changes such as size or length due to external factors, percentage change is a useful concept. Understanding percentage change involves comparing the size of the change to the original value, then converting this comparison into a percentage. To find the percentage change in any scenario where an initial and a final value is given, you can use the formula: \[ \text{Percentage change} = \left( \frac{\text{Change in value}}{\text{Initial value}} \right) \times 100 \% \]This calculation allows us to transform a straightforward numerical change into something more intuitive—a relative change expressed as a percentage. In our exercise, the initial length of the steel track is 25 m, and it changes to 25.01 m. By applying the formula, the percentage change comes out to be 0.04%, which is a tiny increase but significant in understanding material behavior under temperature change.

Change in Length

Change in length is a central concept in thermal expansion, which describes how materials expand or contract with temperature fluctuations. When calculating the change in length, you need two primary measurements: the initial length before the temperature change and the length after this change. The difference between these two values gives us the change in length. Mathematically, this is expressed as: \[ \text{Change in length} = \text{Final length} - \text{Initial length} \]For the steel track example, the initial length is 25 m, and the final length at the increased temperature is 25.01 m. Therefore, the change in length is 0.01 m. This change may seem minimal, but it highlights the importance of accounting for thermal expansion in construction and manufacturing.

• Initial Length: 25 m
• Final Length: 25.01 m
• Change in Length: 0.01 m

Temperature Change

Temperature change is a principal factor driving changes in the physical dimensions of materials. Materials like steel will expand when heated and contract when cooled—a property explained by thermal expansion. In the given exercise, the steel track experiences a temperature increase from 20°C to 70°C. The temperature change, a simple difference between the initial and final temperatures, is crucial as it affects the degree of expansion. For our example, this change is calculated as:\[ \text{Temperature change} = \text{Final temperature} - \text{Initial temperature} \]This results in a temperature change of 50°C for the steel track. Different materials respond uniquely to the same temperature changes due to varying coefficients of thermal expansion, making this calculation important for precise engineering applications.