Physics 2 Exam Question

Temperature and Heat

A copper rod with a length of 2.0 meters and a diameter of 0.02 meters is initially at a temperature of 20°C. The rod is then heated to a final temperature of 80°C. The thermal conductivity of copper is 398 W/(m·K).

a) Calculate the amount of heat transferred through the rod. Show all steps and calculations.

b) If the rod is composed of pure copper (specific heat capacity = 0.39 J/g·°C) and has a mass of 5.0 kg, calculate the change in thermal energy of the rod.

c) Calculate the average thermal conductivity of the rod during the heating process. Show all steps and calculations.

Answer

a) To calculate the amount of heat transferred through the rod, we can use the formula:

Q = k * A * ΔT / L

Where:

• Q is the amount of heat transferred
• k is the thermal conductivity of copper
• A is the cross-sectional area of the rod
• ΔT is the change in temperature
• L is the length of the rod

Given:

• Thermal conductivity of copper, k = 398 W/(m·K)
• Length of the rod, L = 2.0 m
• Initial temperature, T1 = 20°C
• Final temperature, T2 = 80°C
• Diameter of the rod, d = 0.02 m

First, we need to calculate the cross-sectional area of the rod, A:

A = π * (d/2)^2

Substituting the given values:

A = π * (0.02/2)^2 = 0.000314 m^2

Next, we calculate the change in temperature, ΔT:

ΔT = T2 - T1 = 80°C - 20°C = 60°C

Now we can calculate the amount of heat transferred, Q:

Q = k * A * ΔT / L

Q = 398 W/(m·K) * 0.000314 m^2 * 60°C / 2.0 m

Q = 7.4492 W

b) To calculate the change in thermal energy of the rod, we can use the formula:

ΔU = m * C * ΔT

Where:

• ΔU is the change in thermal energy
• m is the mass of the rod
• C is the specific heat capacity of copper
• ΔT is the change in temperature

Given:

• Mass of the rod, m = 5.0 kg
• Specific heat capacity of copper, C = 0.39 J/g·°C
• Change in temperature, ΔT = 60°C

First, we convert the mass from kg to grams:

m = 5.0 kg * 1000 g/kg = 5000 g

Now we can calculate the change in thermal energy, ΔU:

ΔU = m * C * ΔT

ΔU = 5000 g * 0.39 J/g·°C * 60°C

ΔU = 117000 J

c) To calculate the average thermal conductivity of the rod, we can use the formula:

k_avg = Q / (A * ΔT / L)

Where:

• k_avg is the average thermal conductivity
• Q is the amount of heat transferred
• A is the cross-sectional area of the rod
• ΔT is the change in temperature
• L is the length of the rod

Given the values of Q, A, ΔT, and L from parts a):

k_avg = 7.4492 W / (0.000314 m^2 * 60°C / 2.0 m)

k_avg = 23702.5463 W/(m·K)

Therefore, the average thermal conductivity of the rod during the heating process is approximately 23702.55 W/(m·K).