An iron rod has a length of 1000 meters (or 1 km) at 0 degrees Celsius. If the rod must expand 10 cm to bridge a gap, to what temperature must the rod be heated?
To find the temperature to which the iron rod must be heated, we can use the principle of thermal expansion. One commonly used formula for linear expansion is the following:
ΔL = α * L * ΔT
Where:
ΔL = Change in length
α = Coefficient of linear expansion
L = Original length
ΔT = Change in temperature
In this case, we need to find the temperature change (ΔT) that results in a 10 cm (0.1 meters) increase in length for an iron rod with an original length (L) of 1000 meters (1 km).
Step 1: Convert the change in length to meters:
ΔL = 0.1 meters.
Step 2: Plug the known values into the equation:
0.1 = α * 1000 * ΔT
Step 3: Rearrange the equation to solve for ΔT:
ΔT = 0.1 / (α * 1000)
To continue, we need to determine the coefficient of linear expansion (α) for iron. The coefficient of linear expansion for most materials can be found in tables or provided in the problem statement. For iron, the coefficient of linear expansion is typically around 12 x 10^-6 (1/°C).
Step 4: Plug in the value of α:
ΔT = 0.1 / (12 x 10^-6 * 1000)
Simplifying further:
ΔT = 0.1 / 0.012
ΔT = 8.33 °C
Therefore, the iron rod must be heated to approximately 8.33 °C in order to expand by 10 cm and bridge the gap.