The ends of the two rods shown in the figure are separated by 2.97 mm at 24.3 °C. The left-hand rod is brass and 1.21 m long; the right-hand rod is steel and 1.21 m long. Assuming that the outside ends of both rods rest firmly against rigid supports, at what temperature will the ends of the rods that face each other just touch?
Use the following values of linear expansion coefficients for this problem:
αbrass=19 10^−6/°C
αsteel=13 10^−6/°C
To solve this problem, we need to find the temperature at which the ends of the rods touching each other. We can use the concept of linear expansion to solve this.
First, let's calculate the change in length for each rod separately.
For the brass rod:
The length of the brass rod is given as 1.21 m. The coefficient of linear expansion for brass (αbrass) is given as 19 * 10^(-6) / °C.
To find the change in length, we can use the formula:
ΔL = αbrass * L * ΔT
where ΔL is the change in length, αbrass is the coefficient of linear expansion, L is the original length, and ΔT is the change in temperature.
Given that the original separation between the rods is 2.97 mm (which is equal to 0.00297 m), and the change in length of the brass rod is ΔL, we can set up the following equation:
ΔL + ΔL = 0.00297 m
For the steel rod:
The length of the steel rod is also given as 1.21 m. The coefficient of linear expansion for steel (αsteel) is given as 13 * 10^(-6) / °C.
Using the same formula as before, we can calculate the change in length:
ΔL = αsteel * L * ΔT
Now we have two equations:
(1) ΔL + ΔL = 0.00297 m
(2) ΔL = αsteel * L * ΔT
Substituting the values into equation (2):
0.00297 m = (13 * 10^(-6) / °C) * 1.21 m * ΔT
Simplifying the equation:
ΔT = (0.00297 m) / [(13 * 10^(-6) / °C) * 1.21 m]
Evaluating the equation gives us the change in temperature (ΔT).