To determine the constant rate at which the temperature of the aluminum rod must change in order to move the radioactive source at a speed of 96 nm/s, we can use the principles of thermal expansion and the relationship between temperature change and linear displacement. Let's break this down step by step.

Understanding Thermal Expansion

When materials are heated, they expand. The amount of expansion can be quantified using the linear expansion formula:

• ΔL = L₀ * α * ΔT

Where:

• ΔL = change in length
• L₀ = original length of the heated section
• α = coefficient of linear expansion for the material (for aluminum, α ≈ 22 x 10^-6 °C^-1)
• ΔT = change in temperature

Setting Up the Problem

In this scenario, we know:

• The effective heated section of the rod, L₀, is 1.8 cm (or 0.018 m).
• The desired speed of the source, v, is 96 nm/s (or 96 x 10^-9 m/s).

We want to find the rate of temperature change, dT/dt, that corresponds to this speed. First, we need to relate the change in length to the speed of the source.

Relating Speed to Temperature Change

Since the source moves at a constant speed, we can express the speed in terms of the change in length over time:

• v = ΔL / Δt

From the linear expansion formula, we can express ΔL as:

• ΔL = L₀ * α * ΔT

Substituting this into the speed equation gives:

• v = (L₀ * α * ΔT) / Δt

Finding the Rate of Temperature Change

Rearranging the equation to solve for ΔT/Δt yields:

• ΔT/Δt = v / (L₀ * α)

Now we can plug in the values:

• v = 96 x 10^-9 m/s
• L₀ = 0.018 m
• α = 22 x 10^-6 °C^-1

Substituting these values into the equation gives:

• ΔT/Δt = (96 x 10^-9) / (0.018 * 22 x 10^-6)

Calculating the Result

Now, let's perform the calculation:

• ΔT/Δt = (96 x 10^-9) / (0.000396)
• ΔT/Δt ≈ 2.42 x 10^-4 °C/s

This means that to maintain a constant speed of 96 nm/s, the temperature of the aluminum rod must change at a rate of approximately 0.000242 °C per second.

Summary

In summary, by applying the principles of thermal expansion and relating the speed of the radioactive source to the temperature change in the aluminum rod, we found that the temperature must change at a rate of about 0.000242 °C/s to achieve the desired movement speed. This illustrates the fascinating interplay between thermal properties and mechanical motion in materials.