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?

Question

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

Answer 1

total distance= db+ ds

db=1.21*alphabrass*deltatemp
ds=1.21*alphasteel*deltatemp

and total distance=.00297m

solve for delta temp, then add that to 24.3C.

Answer 2

To find the temperature at which the ends of the rods just touch, we need to consider the change in length of both rods due to thermal expansion.

The change in length (∆L) of a material can be calculated using the formula:

∆L = α * L * ∆T

where α is the linear expansion coefficient, L is the original length of the rod, and ∆T is the change in temperature.

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

For the brass rod:
∆L_brass = α_brass * L_brass * ∆T

For the steel rod:
∆L_steel = α_steel * L_steel * ∆T

Since the outside ends of both rods rest firmly against rigid supports, the total change in length must be equal to the initial separation between the ends:

∆L_brass + ∆L_steel = 2.97 mm

Plugging in the values, we have:

(α_brass * L_brass * ∆T) + (α_steel * L_steel * ∆T) = 2.97 mm

Given that α_brass = 19 * 10^(-6)/°C, L_brass = 1.21 m, α_steel = 13 * 10^(-6)/°C, L_steel = 1.21 m, and we need to find ∆T.

So, the equation becomes:

(19 * 10^(-6)/°C * 1.21 m * ∆T) + (13 * 10^(-6)/°C * 1.21 m * ∆T) = 2.97 mm

Simplifying further, we have:

(19 * 10^(-6) + 13 * 10^(-6)) * 1.21 m * ∆T = 2.97 mm

Now, convert ∆T from mm to meters to match the units on the left side of the equation:

2.97 mm = 2.97 x 10^(-3) m

So, the equation becomes:

(19 * 10^(-6) + 13 * 10^(-6)) * 1.21 m * ∆T = 2.97 x 10^(-3) m

Now, solve for ∆T:

∆T = (2.97 x 10^(-3) m) / ((19 * 10^(-6) + 13 * 10^(-6)) * 1.21 m)

Calculating this, we find that ∆T is approximately equal to 86.17 °C.

Therefore, the ends of the rods that face each other will just touch at a temperature of approximately 86.17 °C.