A 5.0-m-diameter solid aluminum sphere is launched into space. By how much does its diameter increase? Give your answer in μm.

Verified step by step guidance

1. Step 1: Identify the physical principle involved. This problem deals with thermal expansion, which describes how materials expand when their temperature increases. The formula for linear expansion is ΔL = α × L₀ × ΔT, where ΔL is the change in length, α is the coefficient of linear expansion, L₀ is the original length, and ΔT is the change in temperature.
2. Step 2: Determine the relevant values. The sphere is made of aluminum, so you need the coefficient of linear expansion for aluminum (α ≈ 23 × 10⁻⁶ /°C). The original diameter of the sphere is L₀ = 5.0 m. You also need the temperature change, ΔT, which should be provided or assumed based on the context of the problem.
3. Step 3: Apply the formula for linear expansion to calculate the change in diameter. Substitute the values into the formula ΔL = α × L₀ × ΔT. Ensure that all units are consistent (e.g., meters for length and °C for temperature).
4. Step 4: Convert the result from meters to micrometers (μm). Since 1 μm = 10⁻⁶ m, multiply the result by 10⁶ to express the change in diameter in micrometers.
5. Step 5: Interpret the result. The calculated value represents the increase in the sphere's diameter due to thermal expansion. Ensure the answer is rounded appropriately and expressed in μm as requested.

Key Concepts

Thermal Expansion

Thermal expansion refers to the increase in size of an object when its temperature rises. In solids, this occurs as the particles vibrate more vigorously and move apart, leading to an increase in dimensions. The degree of expansion is quantified by the coefficient of linear expansion, which varies by material. For aluminum, this coefficient is approximately 23 x 10^-6 /°C.

Coefficient of Linear Expansion

The coefficient of linear expansion is a material-specific constant that quantifies how much a unit length of a material expands per degree of temperature increase. It is essential for calculating the change in dimensions of an object due to temperature changes. For aluminum, this value is crucial for determining how much the diameter of the sphere will increase when subjected to a temperature change.

Conversion of Units

Conversion of units is the process of changing a measurement from one unit to another, which is often necessary in physics problems. In this case, the diameter increase needs to be expressed in micrometers (μm), where 1 meter equals 1,000,000 micrometers. Understanding how to convert between units is essential for providing answers in the required format.