The brass bar and the aluminum bar in the drawing are each attached to an immovable wall. At 29.0 °C the air gap between the rods is 1.93 x 10-3 m. At what temperature will the gap be closed?

To determine the temperature at which the gap between the brass and aluminum bars will be closed, we need to use the concept of thermal expansion.

The formula for linear expansion is given by:
ΔL = α * L0 * ΔT

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

In this case, the gap between the brass and aluminum bars will be closed when the change in length for both bars is equal. So, we can equate the change in length for brass (ΔL_brass) to the change in length for aluminum (ΔL_aluminum).

ΔL_brass = ΔL_aluminum

Using the formula for linear expansion, we can write this as:

α_brass * L0_brass * ΔT_brass = α_aluminum * L0_aluminum * ΔT_aluminum

Now, we know that the original length (L0) for both bars is the same because they are attached to an immovable wall. Therefore, L0_brass = L0_aluminum.

The linear coefficient of expansion for brass (α_brass) and aluminum (α_aluminum) are known values.

To find the change in temperature (ΔT), we can rearrange the equation:

ΔT_brass / α_brass = ΔT_aluminum / α_aluminum

Now, we substitute the given values:
- α_brass = 19 x 10^-6 (linear coefficient of expansion for brass)
- α_aluminum = 23 x 10^-6 (linear coefficient of expansion for aluminum)
- ΔT_brass = T - 29 (change in temperature for brass, where T is the unknown temperature at which the gap is closed)
- ΔT_aluminum = T - 29 (change in temperature for aluminum, where T is the unknown temperature at which the gap is closed)

Now, we can solve the equation:

(ΔT_brass) / (19 x 10^-6) = (ΔT_aluminum) / (23 x 10^-6)

(T - 29) / (19 x 10^-6) = (T - 29) / (23 x 10^-6)

Now, cross-multiplying and simplifying:

23 x 10^-6 * (T - 29) = 19 x 10^-6 * (T - 29)

23T - 667 = 19T - 551

4T = 116

T = 29 + 116/4

T ≈ 58 °C

Therefore, the gap between the brass and aluminum bars will be closed at a temperature of approximately 58 °C.

To solve this problem, we need to consider the thermal expansion of the two bars (brass and aluminum). The gap between the bars will close when the total expansion of the bars is equal to the initial gap of 1.93 x 10^-3 m.

Let's denote:
Lb = initial length of the brass bar
La = initial length of the aluminum bar
ΔT = change in temperature (unknown)

The formula for thermal expansion is given by:
ΔL = α * L * ΔT

Where:
ΔL = change in length
α = coefficient of linear expansion
L = initial length of the bar

The change in length of the brass bar is:
ΔLb = αb * Lb * ΔT

Similarly, the change in length of the aluminum bar is:
ΔLa = αa * La * ΔT

The total change in length will be:
ΔLtotal = ΔLb + ΔLa = αb * Lb * ΔT + αa * La * ΔT

We want to find the temperature at which the total change in length is equal to the initial gap, so we can set up the equation:

ΔLtotal = 1.93 x 10^-3 m

Now, we need to substitute the α values for brass and aluminum. The coefficient of linear expansion for brass is αb = 19 x 10^-6 1/°C, and for aluminum, αa = 23 x 10^-6 1/°C.

Substituting these values into the equation, we get:

(19 x 10^-6 * Lb * ΔT) + (23 x 10^-6 * La * ΔT) = 1.93 x 10^-3 m

We know the lengths and we need to solve for ΔT. Let's assume:
Lb = L0b (initial length of brass bar at 29.0 °C)
La = L0a (initial length of aluminum bar at 29.0 °C)

Let's simplify the equation by factoring out ΔT:
ΔT * (19 x 10^-6 * Lb + 23 x 10^-6 * La) = 1.93 x 10^-3

Now, we can solve for ΔT:

ΔT = (1.93 x 10^-3) / (19 x 10^-6 * Lb + 23 x 10^-6 * La)

Plug in the values for Lb and La to calculate ΔT.