Question

Thermodynamics - Temperature of a metal bar. A metal bar is heated and its temperature is measured every 5 minutes. The table records the temperature readings. \begin{tabular}{lrrrrr} \hline Time (min) & 0 & 5 & 10 & 15 & 20 \\ Temp \(\left({ }^{\circ} \mathrm{C}\right)\) & 21 & 27 & 31 & 33 & 35 \\ \hline \end{tabular} Calculate the rate of increase of temperature in units of \({ }^{\circ} \mathrm{C}\) per minute (a) in the first 5 minutes (b) in the first 10 minutes (c) in the first 15 minutes (d) in the first 20 minutes (e) in the last 5 minutes (f) in the last 10 minutes In each case express your answer as a fraction in its simplest form.

Short answer

Question: Calculate the rate of increase of temperature for the following time frames: a) In the first 5 minutes, b) In the first 10 minutes, c) In the first 15 minutes, d) In the first 20 minutes, e) In the last 5 minutes and f) In the last 10 minutes, given the temperature measurements at different time intervals: Time (min): 0, 5, 10, 15, 20 Temperature: 21, 27, 31, 33, 35

Step-by-step solution

Calculate the temperature differences and time differences

For each time frame, find the difference in temperature and the difference in time. The given measurements are: Time (min): 0, 5, 10, 15, 20 Temperature: 21, 27, 31, 33, 35

Calculate the rate of increase for each time frame

Use the formula: Rate of increase = (Difference in temperature) / (Difference in time). Calculate the rate of increase for each of the specified time frames: (a) In the first 5 minutes: Rate of increase = (27-21)/(5-0) = 6/5 = 1.2 \({ }^\circ \mathrm{C}\) per minute (b) In the first 10 minutes: Rate of increase = (31-21)/(10-0) = 10/10 = 1 \({ }^\circ \mathrm{C}\) per minute (c) In the first 15 minutes: Rate of increase = (33-21)/(15-0) = 12/15 = 4/5 \({ }^\circ \mathrm{C}\) per minute (d) In the first 20 minutes: Rate of increase = (35-21)/(20-0) = 14/20 = 7/10 \({ }^\circ \mathrm{C}\) per minute (e) In the last 5 minutes: Rate of increase = (35-33)/(20-15) = 2/5 = 0.4 \({ }^\circ \mathrm{C}\) per minute (f) In the last 10 minutes: Rate of increase = (35-31)/(20-10) = 4/10 = 2/5 \({ }^\circ \mathrm{C}\) per minute

Key concepts

Temperature Increase Calculation

Understanding how temperature increases over time is an essential concept in thermodynamics and real-world applications, such as in understanding how materials like metals respond to heat. In the given exercise, we explore this by examining temperature data for a heated metal bar at five-minute intervals, requiring us to calculate how much the temperature rises per minute in different time frames.

To perform the temperature increase calculation, we need to subtract the initial temperature from the final temperature for a given period. This gives us the amount of temperature change. We then divide this number by the time interval over which the change occurred to find our rate. For instance, within the first 5 minutes, the metal's temperature rose from 21 to 27 degrees Celsius. The increase is therefore 27°C - 21°C = 6°C, and when we divide by the time (5 minutes), we get a rate of 1.2°C per minute.

Rate of Temperature Change

The rate of temperature change is a way to quantify how quickly the temperature of an object changes over time. It's a crucial concept when analyzing heat transfer or how materials heat up or cool down. In the textbook problem, the rate of temperature change is calculated by using the formula:
\[ \text{Rate of increase} = \frac{\text{Difference in temperature}}{\text{Difference in time}}. \]
Different time intervals exhibit different rates of temperature change, which can tell us more about the heating process. For example, if the rate decreases over time, it might indicate that the bar is approaching a certain temperature where it loses heat to its surroundings as fast as it gains it. Varying rates can also imply specific properties of the metal or its environment, such as thermal conductivity or insulation.

Engineering Mathematics Problems

Solving engineering mathematics problems often involves applying formulas and performing calculated operations — skills that are clearly exemplified in thermodynamics. Thermodynamic calculations, such as the temperature rate of change, require precise mathematical steps and a clear understanding of the underlying physical principles. These types of problems are designed for students to apply arithmetic operations to real-world scenarios, enhancing their analytical and problem-solving skills. Consistent practice on these problems not only solidifies one's grasp of fundamental concepts but also prepares aspiring engineers for the mathematical rigor of their future professions. When approaching engineering mathematics problems, it's important to ensure all units are consistent, to carefully perform each step to avoid errors, and to consider the physical meaning behind the numbers, which aids in comprehending the relationships between the variables in question.