In a certain experiment, it is necessary to be able to move a small radioactive source at selected, extremely slow speeds. This is accomplished by fastening the source to one end of an aluminum rod and heating the central section of the rod in a controlled way. If the effective heated section of the rod is 3.1 cm long, at what constant rate must the temperature of the rod be changed if the source is to move at a constant speed of 26 nm/s?

the formula for thermal expansion is:

[Delta Length]/[initial Length]=k[delta T] so the 1st derivative with respect to time is:

d/dt (L/L0)= k d/dt(T) so
d/dt(T) = L/(L0 k) where k is the expansion coefficient of aluminum.

k = 22.2 m/m Kx10^-6

To determine the constant rate at which the temperature of the rod must change, we can use the equation:

speed = change in length / change in time

In this case, the speed of the source is given as 26 nm/s, and the effective heated section of the rod is 3.1 cm long.

Before proceeding with the calculation, let's convert centimeters to nanometers so that the units are consistent:

1 cm = 10 mm = 10,000 μm = 10,000,000 nm

Therefore, the effective heated section of the rod is:

3.1 cm × 10,000,000 nm/cm = 31,000,000 nm

Now we can calculate the rate of temperature change:

change in length = speed × change in time

31,000,000 nm = 26 nm/s × change in time

To find the change in time, we can rearrange the equation:

change in time = change in length / speed

change in time = 31,000,000 nm / 26 nm/s

Now we can calculate the change in time:

change in time = 1,192,308.5 s

Therefore, the rate at which the temperature of the rod must change to maintain a constant speed of 26 nm/s is approximately:

1,192,308.5 s^(-1) or 1.1923 MHz (megahertz)

To determine the constant rate at which the temperature of the rod must be changed for the source to move at a constant speed of 26 nm/s, we need to consider a few factors.

First, we need to understand the relationship between the temperature change and the resulting linear expansion of the aluminum rod. This relationship is given by the coefficient of linear expansion (α) of the material. The change in length (ΔL) of the rod due to a temperature change (ΔT) can be calculated using the formula:

ΔL = α * L0 * ΔT,

where ΔL is the change in length, α is the coefficient of linear expansion, L0 is the original length of the rod, and ΔT is the temperature change.

Second, we know that the source moves at a constant speed of 26 nm/s. This means that the change in length of the rod should be equal to the speed of the source multiplied by the time it takes for the source to move that distance:

ΔL = speed * time.

By rearranging the equation, we can solve for the time it takes for the source to move the desired distance:

time = ΔL / speed.

Given that the effective heated section of the rod is 3.1 cm long, we need to convert this to meters:

L0 = 3.1 cm = 0.031 m.

Now, we can substitute the values into the equation to calculate the time it takes for the source to move 3.1 cm (or 0.031 m) at a speed of 26 nm/s:

time = 0.031 m / 26 nm/s.

To simplify the calculation, we need to convert nanometers (nm) to meters (m). Since 1 nm is equal to 1 × 10^-9 m, we have:

time = 0.031 m / (26 × 10^-9 m/s).

Dividing the values, we get:

time ≈ 1.192 × 10^6 s.

Therefore, the time it takes for the source to move 3.1 cm at a speed of 26 nm/s is approximately 1.192 × 10^6 seconds.

Finally, to determine the constant rate at which the temperature of the rod must be changed, we divide the change in temperature (ΔT) by the time (t):

rate = ΔT / t.

Since we are looking for a constant rate, we assume that this change occurs over the entire length of the heated section of the rod (3.1 cm or 0.031 m).

However, we cannot calculate the exact temperature change without knowing more about the material properties and the specific heat capacity.