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 use the concept of linear expansion.

The linear expansion of a material is given by the formula:

ΔL = α * L0 * ΔT

Where:
ΔL is the change in length
α is the linear expansion coefficient
L0 is the initial length of the rod
ΔT is the change in temperature

In this case, we need to find the temperature at which the rods just touch, so the change in length should be equal to the initial separation of 2.97 mm or 0.00297 m.

Let's calculate the change in length for both brass and steel rods.

For brass rod:
ΔL_brass = α_brass * L0_brass * ΔT

For steel rod:
ΔL_steel = α_steel * L0_steel * ΔT

Since the two rods are of the same length, we can equate their changes in length:

ΔL_brass = ΔL_steel

α_brass * L0_brass * ΔT = α_steel * L0_steel * ΔT

We can cancel ΔT from both sides:

α_brass * L0_brass = α_steel * L0_steel

Now we can solve for ΔT to find the temperature at which the two rods just touch:

ΔT = (α_brass * L0_brass) / (α_steel * L0_steel)

Substituting the given values:

ΔT = (19 * 10^(-6) / °C * 1.21 m) / (13 * 10^(-6) / °C * 1.21 m)

Simplifying:

ΔT = 19 / 13

Therefore, ΔT = 1.46 °C.

To find the temperature at which the ends of the rods just touch, we need to add this ΔT to the initial temperature of 24.3 °C:

Final temperature = Initial temperature + ΔT
Final temperature = 24.3 °C + 1.46 °C

Thus, the temperature at which the ends of the rods just touch is approximately 25.76 °C.