OII na Sperimentale

1. 95 $\pm$ 0. 06 W Points: 20 Time: 5.0 Hours IPhO 2024 Experiment, English (Official) Page 2 of 21 bundled together with another thermistor ( ) (similar to the other thermistors) have been inserted. These two sensors and the heater are connected to AVA, and AVA indicates the values of their resistance (in ). Figure 4: The smaller copper rod Please observe the following points:
2. Do not touch and activate the buttons on the equipment before it is asked you to do so.

Barra di rame e fori perforati

Vista dal basso della scatola

p.1 — Visione generale apparato sperimentale

Topic: Thermodynamics Metodi: Physical Modeling Competenze: Measurement & Instrumentation Objects: Rod Fonte: Testo (PDF) — p.2 Soluzione: Soluzioni (PDF)

1. 95 $\pm$ 0. 06 W Score: 20 Time: 5.0 hours The following information shall be provided: Experiment, English (Official) Page 2 of 21 bundled together with another thermistor ( ) (similar to the other thermistors) have been Inserted. These two sensors and the heater are connected to AVA, and AVA indicates the values of their resistance (in ). Figure 4: The smaller copper rod Please note the following points:
2. Do not touch and activate the buttons on the equipment before it is asked you to do so.

Copper bar and perforated holes

View from the bottom of the box

The following table shows the results of the experiment:

Topic: Thermodynamics Metodi: Physical Modeling Competenze: Measurement & Instrumentation Objects: Rod Fonte: Testo (PDF) — p.2 Soluzione: Soluzioni (PDF)

to disconnect any wires

2. You do not need to do anything regarding the setup. Take care not to disturb the setup and not to disconnect any wires.

Barra di rame piccola con riscaldatore

Topic: Thermodynamics Metodi: Physical Modeling Competenze: Measurement & Instrumentation Objects: Rod Fonte: Testo (PDF) — p.3 Soluzione: Soluzioni (PDF)

to disconnect any wires

2. You don’t need to do anything regarding the setup. Take care not to disturb the setup and not to disconnect any wires.

Small copper bar with heater

Topic: Thermodynamics Metodi: Physical Modeling Competenze: Measurement & Instrumentation Objects: Rod Fonte: Testo (PDF) — p.3 Soluzione: Soluzioni (PDF)

3. Don’t move the setup during the experiment.

Topic: Thermodynamics Metodi: Physical Modeling Competenze: Measurement & Instrumentation Objects: — Fonte: Testo (PDF) — p.3 Soluzione: Soluzioni (PDF)

3. Don’t move the setup during the experiment.

Topic: Thermodynamics Metodi: Physical Modeling Competenze: Measurement & Instrumentation Objects: — Fonte: Testo (PDF) — p.3 Soluzione: Soluzioni (PDF)

4. If the message “Turn o Heater1” appears on the monitor, immediately turn o Heater no. 1.

Topic: Thermodynamics Metodi: Physical Modeling Competenze: Measurement & Instrumentation Objects: — Fonte: Testo (PDF) — p.3 Soluzione: Soluzioni (PDF)

4. If the message Turn or Heater1 appears on the monitor, immediately turn or Heater no. 1.

Topic: Thermodynamics Metodi: Physical Modeling Competenze: Measurement & Instrumentation Objects: — Fonte: Testo (PDF) — p.3 Soluzione: Soluzioni (PDF)

5. Turning on the heaters will increase the tempereture, so it will take extra time for the system to reach its steady state, make sure that you do not turn on a heater unnecessarily.

Topic: Thermodynamics Metodi: Physical Modeling Competenze: Measurement & Instrumentation Objects: — Fonte: Testo (PDF) — p.3 Soluzione: Soluzioni (PDF)

5. Turning on the heaters will increase the temperature, so it will take extra time for the system to reach its steady state, make sure that you do not turn on a heater unnecessarily.

Topic: Thermodynamics Metodi: Physical Modeling Competenze: Measurement & Instrumentation Objects: — Fonte: Testo (PDF) — p.3 Soluzione: Soluzioni (PDF)

6. Errors need to be calculated and reported whenever the sign is present in the answer sheet.

Topic: Thermodynamics Metodi: Error Propagation Competenze: Error Propagation, Significant Figures Objects: — Fonte: Testo (PDF) — p.3 Soluzione: Soluzioni (PDF)

6. Errors need to be calculated and reported whenever the sign is present in the answer sheet.

Topic: Thermodynamics Metodi: Error Propagation Competenze: Error Propagation, Significant Figures Objects: — Fonte: Testo (PDF) — p.3 Soluzione: Soluzioni (PDF)

7. Regression, (denoted by reg in the formula below and shown as r on the calculator), is a number between 1 and -1 showing how much the data can be tted to a line. If it means that the data are completely on a line. We can use the following formula to calculate the uncertainty of the slope b : in which n is the number of data points. The Equipment R9 $\Omega$ $\pm$ |reg| = 1 $\text{Deltab}$ = $b\sqrt{}$ 1 $(n-2$) ( 1 reg2 $-1$) Points: 20 Time: 5.0 Hours IPhO 2024 Experiment, English (Official) Page 3 of 21

Topic: Thermodynamics Metodi: Experimental Data Analysis, Graph Linearization Competenze: Graph Linearization, Experimental Data Analysis, Error Propagation Objects: — Fonte: Testo (PDF) — p.3 Soluzione: Soluzioni (PDF)

7. Regression, (denoted by reg in the formula below and shown as r on the calculator), is a number between 1 and -1 showing how much the data can be tted to a line. If It means that The data is completely on line. We can use the following formula to calculate the uncertainty of the slope b: where n is the number of data points. The Equipment R9 $\Omega$ $\pm$ The value of the product is 1 $\text{Deltab}$ = $b\sqrt{}$ 1 $(n-2$) ( 1 reg2 $-1$) Score: 20 Time: 5.0 hours The following information shall be provided: Experiment, English (Official) Page 3 of 21

Topic: Thermodynamics Metodi: Experimental Data Analysis, Graph Linearization Competenze: Graph Linearization, Experimental Data Analysis, Error Propagation Objects: — Fonte: Testo (PDF) — p.3 Soluzione: Soluzioni (PDF)

1. Generally, Electrical resistors have di erent behaviors in response to a change in temperature. One of the widely used resistors is PT100 which has a linear behavior over a considerable range of temperatures, i.e. (1) in which is the value of the resistance at the , is a constant coe cient (within the range of temperatures for this problem), and is the temperature in degrees Celsius. The value of for the resistor used in this problem is . For PT100, .

Topic: Thermodynamics, Circuits Metodi: Physical Modeling, Experimental Data Analysis Competenze: Mathematical Modeling, Graph Linearization Objects: Resistor Fonte: Testo (PDF) — p.4 Soluzione: Soluzioni (PDF)

1. Generally, electrical resistors have erent behaviors in response to a change in temperature. One of the widely used resistors is PT100 which has a linear behavior over a considerable range of The following table shows the results of the calculations: (1) in which is the value of the resistance at the , is a constant coe The following table shows the number of samples taken: And the temperature is in degrees Celsius. The value of for the resistor used in this problem is . For PT100, .

Topic: Thermodynamics, Circuits Metodi: Physical Modeling, Experimental Data Analysis Competenze: Mathematical Modeling, Graph Linearization Objects: Resistor Fonte: Testo (PDF) — p.4 Soluzione: Soluzioni (PDF)

2. The thermistors 1 through 9 have a nonlinear behavior in response to changes in temperature, and are usually used for measuring small changes in temperature. The resistance of these thermistors changes with temperature as follows: (2) Where is a constant, is the Boltzmann constant, is the temperature in kelvins and is the energy gap for the thermistor’s semiconductor material. Remember that

Topic: Thermodynamics, Kinetic Theory Metodi: Physical Modeling, Graph Linearization Competenze: Mathematical Modeling, Graph Linearization Objects: Resistor Fonte: Testo (PDF) — p.4 Soluzione: Soluzioni (PDF)

2. The thermistors 1 through 9 have nonlinear behavior in response to temperature changes, and are usually used for measuring small changes in temperature. The resistance of these thermistors changes with temperature as follows: (2) Where is a constant, is the Boltzmann constant, is the temperature in kelvins and is the energy gap for the thermistors semiconductor material. Remember that

Topic: Thermodynamics, Kinetic Theory Metodi: Physical Modeling, Graph Linearization Competenze: Mathematical Modeling, Graph Linearization Objects: Resistor Fonte: Testo (PDF) — p.4 Soluzione: Soluzioni (PDF)

3. The digital device AVA was designed by Iranian engineers speci cally for this experiment. AVA measures the instantaneous resistances of Thermistors 1 through 7, PT100, and or . every two seconds. Then, based on the formula 2, reports the temperature of these sensors. However, for Sensor no. 9, only the value of the resistance is displayed. On the left of AVA’s monitor, the temperatures of sensors 1 through 7 are displayed in a column. The last row, however, only shows at each instance, either the temperature of sensor 8, or the resistance of Sensor 9 ( ). You can toggle between these two values by pressing the button shown in Figure 5. AVA also has a timer, which works very much like any commercial timer: by pressing the Start/Stop button, the timer starts measuring the time elapsed, and pressing the Start/Stop button again stops the timer. While the timer is working, pressing the Lap button will result in all the displayable values being saved. Pressing the same button when the timer is not working resets the timer, however, the saved data will not be erased. To erase the data the Lap/Reset button has to pressed and held for 5 seconds. The saved data can be seen by repeatedly pressing the Next/Prev button: pressing Prev shows older data, pressing Next shows newer data. All saved data will be erased in case the device is switched o and on. The device will not turn o automatically. R = R0(1 + $\alpha \theta$) R0 $0^{\circ }C\alpha$ $\theta$ $\alpha$
4. $0039083^{\circ }C-1$ R0 = 100. 00 $\Omega$ $R^{\prime }$ = $R^{\prime }0e$ Eg 2kBT $R{0}^{\prime }$ kB = 8. 61733 $\times 10-5$ eV /K T Eg T = $(\theta$ + 273. 15) K $\text{thetab}$ R9 R9 $\text{thetab}-R9$ Points: 20 Time: 5.0 Hours IPhO 2024 Experiment, English (Official) Page 4 of 21 Figure 5: Start/Stop and Lap buttons. (Further details of keys’ functions come as point 3, on the page 4.)

Topic: Thermodynamics Metodi: Physical Modeling, Experimental Data Analysis Competenze: Measurement & Instrumentation, Experimental Data Analysis Objects: Resistor Fonte: Testo (PDF) — p.4 Soluzione: Soluzioni (PDF)

3. The digital device AVA was designed by Iranian engineers specifically for this experiment. AVA Measures the instantaneous resistances of Thermistors 1 through 7, PT100, and or . every Two seconds. Then, based on formula 2, reports the temperature of these sensors. However, for Sensor is off. 9, only the value of the resistance is displayed. On the left of AVAs monitor, the temperatures of sensors 1 through 7 are displayed in a column. The last row, however, only shows at each instance, either the temperature of sensor 8, or the resistance of sensor 9 ( ). You can toggle between these two values by pressing the button shown in Figure 5. AVA also has a timer, which works very much like any commercial timer: by pressing the Start/Stop button, the timer starts measuring the time elapsed, and pressing the Start/Stop button again stops the timer. While the timer is working, pressing the Lap button will result in all the displayable values being saved. Pressing the same button when the timer is not working resets the timer, However, the saved data will not be erased. To erase the date the Lap/Reset button has to be pressed And held for 5 seconds. The saved data can be seen by repeatedly pressing the Next/Prev button: pressing Prev shows old date, pressing next shows newer date. All saved data will be erased in case the device is switched or on. The device will not turn or automatically. R = R0(1 + $\alpha \theta$) R0 $0^{\circ }C\alpha$ $\theta$ $\alpha$
4. $0039083^{\circ }C-1$ R0 = 100. 00 $\Omega$ $R^{\prime }$ = $R^{\prime }0e$ Eg 2kBT $R{0}^{\prime }$ kB = 8. 61733 $\times 10-5$ eV /K T Eg T = $(\theta$ + 273. 15) K $\text{thetab}$ R9 R9 $\text{thetab}-R9$ Score: 20 Time: 5.0 hours The following information shall be provided: Experiment, English (Official) Page 4 of 21 Figure 5: Start/Stop and Lap buttons. (Further details of keys’ functions as point 3, on page 4.)

Topic: Thermodynamics Metodi: Physical Modeling, Experimental Data Analysis Competenze: Measurement & Instrumentation, Experimental Data Analysis Objects: Resistor Fonte: Testo (PDF) — p.4 Soluzione: Soluzioni (PDF)

4. As shown in Figure 5, there is one switch for turning the fans on and o ( ) and there are also three switches for turning on the heater ( ). The icon for each switch is stamped below it. The rst heater is a 1.95 W heater, the power of second heater is written on the device as shown in Figure 6, and the power of third heater is unknown. Points: 20 Time: 5.0 Hours IPhO 2024 Experiment, English (Official) Page 5 of 21 Figure 6: The power of second heater Theory In this problem, heat is transferred inside the rod through conduction, and transferred from the rod to the surrounding air through natural or forced convection. Also, due to the heat capacity of the rod, some of the heat injected into the rod is used up to raise the temperature of the rod. (a) Heat conduction: for a heat conductor in the shape of a rod with no heat loss from its lateral surface, the rate of heat transfer, , through a di erential element (Figure 7) at the steady state is as follows (3) where is the temperature di erence between the two edges of the di erential element, is the cross-sectional area, is the length of the di erential element, and k is the heat transfer coe cient (heat conductivity) which depends on the type of material the rod is made of. dQ/dt dQ dt = $-kAd\theta$ dx $d\theta$ A dx Points: 20 Time: 5.0 Hours IPhO 2024 Experiment, English (Official) Page 6 of 21 Figure 7: The di erential heat conductor element. (heat conduction, in the absence of convection) (b) Convection: For any object exchanging heat with the air through its lateral surface, the following relationship holds: (4) in which is the area of the lateral surface, is the temperature di erence between the object and the surrounding air, and is the convective heat transfer coe cient which is a function of the shape of the object and the nature of the heat ow through the sides of the object. The Experiment: In order to save time, we recommend that you turn on Heater 2 and the fans which are needed in Part B. Make sure that the other heaters are not turned on. Part A: The short copper rod (3.9 points) A-0 Write numbers 0 to 9 in the table. 0 pt Before turning on Heater 1, the small rod is at the same temperature as its environment. Tasks A-1 to A-3 are related to the heating process and tasks A-4 to A-7 correspond to the cooling process of the rod. A-1 Record the initial value of resistance (resistance of PT100) when it is at the temperature of its environment. Using Eq.1, nd this temperature. 0.2 pt Let us denote the total heat capacity of the rod and the heater and the sensors by . To nd we should turn on Heater 1 and measure the change in the value of resistance for at least 150 seconds. Note that there is a time delay in the heating and cooling of the sensors. dQ dt = $-hS\Delta \theta$ S $\Delta \theta$ h Renv CS CS Points: 20 Time: 5.0 Hours IPhO 2024 Experiment, English (Official) Page 7 of 21 A-2 In time intervals of approximately 10 seconds record the value of . Do this at least 15 times. 0.5 pt A-3 Draw a diagram for Part A-2, t a line to your data and nd its slope. Using the slope nd . 0.8 pt Wait until the value of reaches and then turn o Heater 1. The temperature of the rod and the resistance will start to decrease slowly after a few seconds. When PT100 is cooling down, the resistance is given by: (5) in which and are constants. A-4 At several di erent instances of time, measure the value of . 0.5 pt A-5 Make a semi-logarithmic plot of your data in part A-4. Then nd . 0.7 pt The insulator around the rod causes the resistors to reach thermal equilibrium sooner, and, the rod to have a more uniform temperature pro le. A-6 Measure and record the resistance of Thermistor 9 in terms of . Measure at least seven di erent values of , preferably in the range. 0.5 pt A-7 Draw a diagram of the resistance versus on the given semilogarithmic graph and nd the magnitude of the energy gap in units of eV. 0.7 pt Part B: The long copper rod (4.1 points) Wait for the rod to reach a steady state i.e. the measured temperatures at all points remain constant, and then answer the following questions. We shall denote the temperatures of Thermistors 1-7 by through respectively. The location of Thermistor 1 corresponds to . B-1 When the rod reaches steady state, measure and record the temperature and for the seven points on the rod. 0.4 pt R CS R 120 $\Omega$ R R $-Renv$ = $Ae-\text{gammat}$ A $\gamma$ R $-Renv$ $\gamma$ R9 R R 114 $-120\Omega$ R9 1/T Eg $\theta 1$ $\theta 7$ x = 0 $\text{thetab}$ $\text{thetax}$ Points: 20 Time: 5.0 Hours IPhO 2024 Experiment, English (Official) Page 8 of 21 In order to save time, take a look at Part C and then continue Part B. B-2 On semi-log paper, draw a diagram for the di erence between the temperature of the box and the temperature at point , along the length of the rod. 0.4 pt It can be shown that as a function of the distance from Heater 2 along the length of the line, the temperature obeys the following relation: (6) in which the ambient temperature of the box, and are constants, and where is the radius of the rod. As a rst step, we can ignore the data corresponding to large values of and take to be zero. In this case we can nd and to a rst approximation, let us call them and : B-3 Use the temperatures through and nd and using the diagram of Part B-2. 0.6 pt The temperature at the end of the rod farthest from the heater does not change with . Assume this happens around a distance . One can use this to determine in terms of , , and : B-4 Express in terms of , , and , and for . Find its numerical value using the results of Part B-3. Denote this quantity as . 0.4 pt Now we can use the value obtained for to correct the previous calculation. To do so, assume: B-5 Find through and complete the columns added to table B-1. 0.4 pt B-6 Draw a new diagram to obtain the values for and Denote them by and respectively. 1.0 pt To obtain accurate approximations, the corrections should be repeated many times, but in the end, we’ll nd that the nal answer is close to and . B-7 By balancing the input and output powers of the copper rod, nd and . 0.9 pt x $(\text{thetax}-\text{thetab}$) $\text{thetax}$ = $\text{thetab}$ + $Ae-\text{lambdax}$ + $Be\text{lambdax}$ $\text{thetab}$ A B $\lambda$ = $\sqrt{}2h$ kr r x B $\lambda$ A $\lambda (0$) A(0) $\theta 1$ $\theta 5$ $\lambda (0$) A(0) x x = d B $\lambda$ A d B $\lambda$ A d d = 44. 0 cm B(1) B $\text{thetax}$ ${}^{\prime }$ = $\text{thetax}-B(1)e\lambda (0)x$ $\theta 1$ ${}^{\prime }$ $\theta 7$ ${}^{\prime }$ $\lambda$ A $\lambda (1$) A(1) $\lambda$ = $\lambda (0)+\lambda (1$) 2 A = A(0)+A(1) 2 h k Points: 20 Time: 5.0 Hours IPhO 2024 Experiment, English (Official) Page 9 of 21 Part C: Measuring the unknown power (2.0 points) While the fans are on, turn on Heaters 2 and 3, and wait until the temperature reaches equilibrium at all points of the copper rod. (This will take about 15 minutes.) C-1 Measure and record the temperatures through . 0.4 pt It can be shown that, in this case, the temperature in terms of varies as follows in which is a constant and is the hyperbolic cosine function of u de ned as: C-2 Draw the diagram of temperature versus distance and nd . 0.6 pt C-3 Using your own method to nd the e ective power of Heater 3. Explain your method as clearly as possible by writing down explicitly the mathematical formulas you have used to arrive at your results. 1.0 pt $\theta 1$ $\theta 7$ x $\text{thetax}-\text{thetab}$ = $A^{\prime }cosh(\lambda (x-x0$)) $A^{\prime }$ cos h(u) cos h(u) = $eu+e-u$ 2 x0 Points: 20 Time: 5.0 Hours IPhO 2024 Experiment, English (Official) Page 10 of 21

Pannello AVA con tasti Start/Stop e Lap

Potenza del secondo riscaldatore

p.7 — Elemento differenziale conduttore di calore

Topic: Thermodynamics, Elasticity & Materials Metodi: First Law of Thermodynamics, Differential Equations, Experimental Data Analysis, Graph Linearization Competenze: Mathematical Modeling, Graph Linearization, Experimental Data Analysis Objects: Rod Fonte: Testo (PDF) — p.5 Soluzione: Soluzioni (PDF)

4. As shown in Figure 5, there is one switch for turning the fans on and o ( ) and there are also three switches for turning on the heater ( ). The icon for each switch is printed below it. The rst heater is a 1.95 W heater, the power of second heater is written on the device as shown in Figure 6, and the power of third heater is unknown. Score: 20 Time: 5.0 hours The following information shall be provided: Experiment, English (Official) Page 5 of 21 Figure 6: The power of second heater Theory In this problem, heat is transferred inside the rod through conduction, and transferred from the rod to the surrounding air through natural or forced convection. Also, due to the heat capacity of The rod, some of the heat injected into the rod is used up to raise the temperature of the rod. (a) Heat conduction: for a heat conductor in the shape of a rod with no heat loss from its side surface, the rate of heat transfer, , through a di erential element (Figure 7) at the steady state is as follows (3) where is the temperature of erence between the two edges of the di erential element, is the cross-sectional area, is the length of the di erential element, and k is the heat transfer Other Heat conductivity which depends on the type of material the rod is made of. dQ/dt dQ dt = $-kAd\theta$ dx $d\theta$ A dx Score: 20 Time: 5.0 hours The following information shall be provided: Experiment, English (Official) Page 6 of 21 Figure 7: The element of the heat conductor. (heat conduction, in the absence of convection) (b) Convection: For any object exchanging heat with the air through its lateral surface, the following relationship holds: (4) in which is the area of the lateral surface, is the temperature of erence between the object and the surrounding air, and is the convective heat transfer coe cent which is a function of the The shape of the object and the nature of the heat through the sides of the object. The Experiment: In order to save time, we recommend that you turn on Heater 2 and the fans which are needed in Part B. Make sure the other heaters are not turned on. Part A: The short copper rod (3.9 points) A-0 Write numbers 0 to 9 in the table. 0 pt Before turning on Heater 1, the small rod is at the same temperature as its environment. The following tasks are to be performed: A-4 to A-7 correspond to the cooling process of the heating process. The rod. A-1 Record the initial value of resistance (resistance of PT100) when it is at the temperature of its environment. Using Eq.1, nd this temperature. 0.2 pt Let us denote the total heat capacity of the rod and the heater and the sensors by . To nd We should turn on Heater 1 and measure the change in the resistance value for at least 150 Seconds. Note that there is a time delay in the heating and cooling of the sensors. dQ dt = $-hS\Delta \theta$ S $\Delta \theta$ h Other CS CS Score: 20 Time: 5.0 hours The following information shall be provided: Experiment, English (Official) Page 7 of 21 A-2 In time intervals of approximately 10 seconds record the value of . Do This is at least 15 times. 0.5 pt A-3 Draw a diagram for Part A-2, t a line to your data and nd its slope. Using the slope nd . 0.8 pt Wait until the value of reaches and then turn or Heater 1. The temperature of the rod And the resistance will start to decrease slowly after a few seconds. When PT100 is cooling down, the resistance is given by: (5) in which and are constants. A-4 At several of the most recent instances of time, measure the value of . 0.5 pt A-5 Make a semi-logarithmic plot of your data in part A-4. Then don ‘t . 0.7 pt The insulator around the rod causes the resistors to reach thermal equilibrium sooner, and, the The Commission has also proposed that the measures be taken to ensure that the A-6 Measure and record the resistance of Thermistor 9 in terms of . Measure at least seven of the erent values of , preferably in the The range. 0.5 pt A-7 Draw a diagram of the resistance Other on the given semilogarithmic graph and nd the magnitude of the energy gap in units of eV. 0.7 pt Part B: The long copper rod (4.1 points) Wait for the rod to reach a steady state i.e. the measured temperatures at all points remain constant, and then answer the following questions. We shall indicate the temperatures of Thermistors 1-7 by through The Commission shall adopt the following measures: The location of Thermistor 1 corresponds to . B-1 When the rod reaches steady state, measure and record the temperature and For the seven points on the rod. 0.4 pt R CS R 120 $\Omega$ R R $-Renv$ = $Ae-\text{gammat}$ A $\gamma$ R $-Renv$ $\gamma$ R9 R R 114 $-120\Omega$ R9 1/T Eg $\theta 1$ $\theta 7$ x = 0 $\text{thetab}$ $\text{thetax}$ Score: 20 Time: 5.0 hours The following information shall be provided: Experiment, English (Official) Page 8 of 21 In order to save time, take a look at Part C and then continue Part B. B-2 On semi-log paper, draw a diagram for the di erence between the temperature of the box and the temperature at point , along the length of the rod. 0.4 pt It can be shown that as a function of the distance from Heater 2 along the length of the line, the temperature obeys the following relation: (6) in which the ambient temperature of the box, and are constants, and where is the radius of the rod. As a rst step, we can ignore the data corresponding to large values of and take to be zero. In this case we can and to a rst approximation, let us call them and : B-3 Use the temperatures through and nd and using the diagram of Part B-2. 0.6 pt The temperature at the end of the rod farthest from the heater does not change with . He assumes This happens around a distance. . One can use this to determine in terms of , , and: B-4 Express in terms of , , and , and for . Find its numerical value using the results of Part B-3. Denote this quantity as . 0.4 pt Now we can use the value obtained for to correct the previous calculation. To do so, he assumes: B-5 Find through and complete the columns added to table B-1. 0.4 pt B-6 Draw a new diagram to obtain the values for and Denote them by and The Commission shall adopt the following measures: 1.0 pt To obtain accurate approximations, the corrections should be repeated many times, but at the end, We’ll find that the nal answer is close to and . B-7 By balancing the input and output powers of the copper rod, nd and . 0.9 pt x $(\text{thetax}-\text{thetab}$) $\text{thetax}$ = $\text{thetab}$ + $Ae-\text{lambdax}$ + $Be\text{lambdax}$ $\text{thetab}$ A B $\lambda$ = $\sqrt{}2h$ kr r x B $\lambda$ A $\lambda (0$) A(0) $\theta 1$ $\theta 5$ $\lambda (0$) A(0) x x = d B $\lambda$ A d B $\lambda$ A d d = 44. 0 cm B(1) B $\text{thetax}$ ${}^{\prime }$ = $\text{thetax}-B(1)e\lambda (0)x$ $\theta 1$ ${}^{\prime }$ $\theta 7$ ${}^{\prime }$ $\lambda$ A $\lambda (1$) A(1) $\lambda$ = $\lambda (0)+\lambda (1$) 2 A = A(0)+A(1) 2 h k Score: 20 Time: 5.0 hours The following information shall be provided: Experiment, English (Official) Page 9 of 21 Part C: Measuring the unknown power (2.0 points) While the fans are on, turn on Heaters 2 and 3, and wait until the temperature reaches equilibrium. At all points of the copper rod. (This will take about 15 minutes.) C-1 Measure and record the temperatures through . 0.4 pt It can be shown that, in this case, the temperature in terms of varies as follows in which is a constant and is the hyperbolic cosine function of u de ned as: C-2 Draw the diagram of temperature versus distance and nd . 0.6 pt C-3 Using your own method to nd the ective power of Heater 3. Explain Your method as clearly as possible by writing down explicitly the mathematical formulas you have used to arrive at your results. 1.0 pt $\theta 1$ $\theta 7$ x $\text{thetax}-\text{thetab}$ = $A^{\prime }cosh(\lambda (x-x0$)) $A^{\prime }$ (i) the following information is provided: cos h(u) = $eu+e-u$ 2 x0 Score: 20 Time: 5.0 hours The following information shall be provided: Experiment, English (Official) Page 10 of 21

Pannello AVA con tasti Start/Stop e Lap

Power of the second heater

p.7 — Elemento differenziale conduttore di calore

Topic: Thermodynamics, Elasticity & Materials Metodi: First Law of Thermodynamics, Differential Equations, Experimental Data Analysis, Graph Linearization Competenze: Mathematical Modeling, Graph Linearization, Experimental Data Analysis Objects: Rod Fonte: Testo (PDF) — p.5 Soluzione: Soluzioni (PDF)