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 find the temperature at which the ends of the rods just touch, we need to equate the change in length of the brass rod with the change in length of the steel rod.
The change in length for a rod can be calculated using the formula:
ΔL = α * L * ΔT
Where:
ΔL is the change in length
α is the linear expansion coefficient
L is the original length of the rod
ΔT is the change in temperature
Let's start by calculating the change in length for both the brass and steel rods.
For the brass rod:
ΔLbrass = αbrass * Lbrass * ΔT
For the steel rod:
ΔLsteel = αsteel * Lsteel * ΔT
Since the ends of the rods just touch, the total change in length of both rods should be equal.
Therefore, we can set up the equation:
αbrass * Lbrass * ΔT = αsteel * Lsteel * ΔT
We can cancel out ΔT from both sides of the equation:
αbrass * Lbrass = αsteel * Lsteel
Now we can substitute the given values:
19 * 10^-6 / °C * 1.21 m = 13 * 10^-6 / °C * 1.21 m
Simplifying the equation further:
22.99 * 10^-6 / °C = 15.73 * 10^-6 / °C
Dividing both sides by 10^-6, we get:
22.99 * 10 = 15.73 * 10
Now divide both sides by 10 to isolate the term:
22.99 = 15.73
This equation is not true, which means the two rods will never touch each other.
Therefore, there is no temperature at which the ends of the rods will just touch.