AP Physics 2 Exam Question

A metal rod of length 1 meter and cross-sectional area 0.01 square meters is initially at a temperature of 20°C. One end of the rod is placed in contact with a heat source at a temperature of 100°C, while the other end is in contact with a heat sink at a temperature of 0°C. The rod is made of a material with a thermal conductivity of 200 W/(m·°C) and a thermal resistance of 0.1 (°C/W).

a) Calculate the rate at which heat flows through the rod. b) Determine the temperature at the midpoint of the rod after 10 seconds. c) Calculate the total heat transferred through the rod after 10 seconds.

Assume steady-state conditions and neglect any heat loss to the surroundings.

Answer with Step-by-Step Explanation

a) To calculate the rate at which heat flows through the rod, we need to use Fourier's Law of Heat Conduction:

Q = -k * (A / L) * ΔT

Where:

• Q is the rate of heat flow (in watts),
• k is the thermal conductivity of the material (in watts per meter per degree Celsius),
• A is the cross-sectional area of the rod (in square meters),
• L is the length of the rod (in meters),
• ΔT is the temperature difference between the heat source and the heat sink (in degrees Celsius).

Plugging in the given values:

Q = -200 * (0.01 / 1) * (100 - 0)
Q = -200 W

Therefore, the rate at which heat flows through the rod is 200 watts.

b) To determine the temperature at the midpoint of the rod after 10 seconds, we need to use the thermal resistance equation:

ΔT = Q * R

Where:

• ΔT is the temperature difference (in degrees Celsius),
• Q is the rate of heat flow (in watts),
• R is the thermal resistance (in degrees Celsius per watt).

Plugging in the given values:

ΔT = 200 * 0.1
ΔT = 20°C

Since the temperature at the heat source is 100°C, the temperature at the midpoint of the rod after 10 seconds will be:

T_midpoint = 100 - 20
T_midpoint = 80°C

Therefore, the temperature at the midpoint of the rod after 10 seconds is 80°C.

c) To calculate the total heat transferred through the rod after 10 seconds, we need to use the formula:

Q_total = Q * t

Where:

• Q_total is the total heat transferred (in joules),
• Q is the rate of heat flow (in watts),
• t is the time duration (in seconds).

Plugging in the given values:

Q_total = 200 * 10
Q_total = 2000 J

Therefore, the total heat transferred through the rod after 10 seconds is 2000 joules.