AP Physics 2 Exam Question

Three identical rods, A, B, and C, made of different materials are connected together in series, as shown in the diagram below. Rod A has a thermal conductivity of 100 W/m·K, rod B has a thermal conductivity of 200 W/m·K, and rod C has a thermal conductivity of 50 W/m·K. The lengths of the rods are 2 m, 3 m, and 1 m, respectively. The first end of the rod A is placed in thermal contact with a heat source at a temperature of 100°C, while the other end of rod C is placed in thermal contact with a heat sink at a temperature of 0°C.

a) Calculate the total thermal resistance of the three rods connected in series.

b) If the heat source supplies a power of 500 W, determine the temperature at the junction between rod A and rod B.

c) Calculate the heat flux through each rod.

Answer

a) The total thermal resistance of a series connection of rods is given by the sum of the thermal resistances of each individual rod. The thermal resistance of a rod is given by the formula:

R = L / (k * A)

Where:

• R is the thermal resistance,
• L is the length of the rod,
• k is the thermal conductivity of the material, and
• A is the cross-sectional area of the rod.

For rod A: R_A = (2 m) / (100 W/m·K * (A_A))

For rod B: R_B = (3 m) / (200 W/m·K * (A_B))

For rod C: R_C = (1 m) / (50 W/m·K * (A_C))

The total thermal resistance of the series connection is the sum of these individual thermal resistances:

R_total = R_A + R_B + R_C

b) The temperature at the junction between rod A and rod B can be determined using the equation for steady-state heat conduction:

Q = h * A * (T_hot - T_cold)

Where:

• Q is the heat transfer rate,
• h is the heat transfer coefficient,
• A is the cross-sectional area of the junction, and
• T_hot and T_cold are the temperatures of the hot and cold sides, respectively.

Since the rods are in series, the heat transfer rate through each rod is the same:

Q_A = Q_B = Q_C = 500 W

Using the thermal resistance formula again, we can calculate the temperature difference across each rod:

dT_A = Q_A * R_A dT_B = Q_B * R_B dT_C = Q_C * R_C

The temperature at the junction between rod A and rod B is:

T_junction = T_hot - dT_A + dT_B

c) The heat flux through each rod can be calculated using Fourier's law of heat conduction:

Q = k * A * dT / L

For rod A: Q_A = k_A * A_A * dT_A / L_A

For rod B: Q_B = k_B * A_B * dT_B / L_B

For rod C: Q_C = k_C * A_C * dT_C / L_C

This question assesses the student's understanding of thermal resistance, heat transfer equations, and thermal conductivity.