Textbook Question
An iron cube floats in a bowl of liquid mercury at 0°C. By what percent will the fraction of volume submerged change? [Hint: See Chapter 13.]
286
views
20. Heat and Temperature
Linear Thermal Expansion
5:20 minutesProblem 13
Textbook Question
(II) To make a secure fit, rivets that are larger than the rivet hole are often used and the rivet is cooled (usually in dry ice) before it is placed in the hole. A steel rivet 1.872 cm in diameter is to be placed in a hole 1.870 cm in diameter in a metal at 24°C. To what temperature must the rivet be cooled if it is to fit in the hole?
Verified step by step guidance1
Determine the relationship between the change in temperature and the change in diameter of the steel rivet using the formula for linear thermal expansion: ΔL = αL₀ΔT, where ΔL is the change in length (or diameter in this case), α is the coefficient of linear expansion for steel, L₀ is the original diameter, and ΔT is the change in temperature.
Calculate the required change in diameter (ΔL) for the rivet to fit into the hole. This is given by the difference between the hole's diameter and the rivet's original diameter: ΔL = 1.870 cm - 1.872 cm.
Rearrange the linear thermal expansion formula to solve for the change in temperature (ΔT): ΔT = ΔL / (αL₀). Substitute the values for ΔL, α (the coefficient of linear expansion for steel, typically around 12 × 10⁻⁶ /°C), and L₀ (1.872 cm).
Determine the final temperature (T_final) by subtracting the change in temperature (ΔT) from the initial temperature (T_initial = 24°C): T_final = T_initial - ΔT.
Verify the result conceptually by ensuring that the rivet's diameter at the calculated temperature is less than or equal to the hole's diameter, confirming that it will fit securely.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
5mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Thermal Expansion
Thermal expansion refers to the tendency of materials to change their dimensions in response to temperature changes. As temperature increases, most materials expand, leading to an increase in size. Conversely, cooling a material typically causes it to contract. Understanding this concept is crucial for determining how much a rivet will shrink when cooled, allowing it to fit into a smaller hole.
Recommended video:
Guided course
05:21
Volume Thermal Expansion
Coefficient of Linear Expansion
The coefficient of linear expansion is a material-specific value that quantifies how much a unit length of a material expands or contracts per degree change in temperature. It is essential for calculating the change in diameter of the rivet as it is cooled. Different materials have different coefficients, which must be considered to accurately predict the rivet's behavior when subjected to temperature changes.
Recommended video:
Guided course
07:31Linear Thermal Expansion
Temperature Conversion
Temperature conversion is the process of changing a temperature value from one scale to another, such as Celsius to Kelvin. In physics problems involving thermal expansion, it is often necessary to convert temperatures to ensure consistency in calculations. Understanding how to convert temperatures accurately is vital for determining the final temperature to which the rivet must be cooled to fit into the hole.
Recommended video:
Guided course
07:46Unit Conversions
7:31mWatch next
Master Linear Thermal Expansion with a bite sized video explanation from Patrick
Start learningRelated Videos
Related Practice