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Chapter 1: Problem 6

One method of getting a tight fit, say of a metal peg in a hole in a metal block, is to manufacture the peg slightly larger than the hole. The peg is then inserted when at a different temperature than the block. Should the block be hotter or colder than the peg during insertion? Explain your answer.

Short Answer

Expert verified
The metal block should be hotter than the peg during insertion. This is because heating the block will cause the hole to expand, allowing the peg, which is at a lower temperature, to be inserted. As the temperatures of the peg and the block equalize, the peg will expand and fill the hole tightly, ensuring a snug fit.

Step by step solution

01

Understand the concept of thermal expansion

Thermal expansion occurs when the temperature of an object changes, causing it to expand or contract. As the temperature increases, particles gain energy and move faster, leading to an increase in the size of the object. On the other hand, as the temperature decreases, particles lose energy and move slower, leading to a contraction in the size of the object.
02

Determine the desired outcome

We want to insert a metal peg into a hole in a metal block with a tight fit, meaning that there should be very little or no space between the peg and the hole when they are at the same temperature. In order to achieve this tight fit, we need to ensure that either the peg becomes smaller when its temperature is changed or the hole in the block becomes larger when its temperature is changed.
03

Analyze the temperature relationship

We are given that the metal peg is slightly larger than the hole in the metal block. Our goal is to find a temperature difference between the peg and the block that will allow the peg to be inserted into the hole. When the peg is inserted into the hole, we want its diameter to be smaller or equal to the hole's diameter for the initial insertion process. There are two possible scenarios, which we will analyze: Scenario 1: We heat the block, which causes the hole to expand. The peg remains at a lower temperature. We need to see whether this increase in the hole diameter will allow the peg to be inserted. Scenario 2: We cool the peg, which causes it to contract. The block remains at a higher temperature. We need to see whether this decrease in the peg diameter will allow it to be inserted into the hole.
04

Determine the best option

For Scenario 1: Heating the block expands the hole, allowing the peg, which is at a lower temperature, to be inserted into the larger hole. After insertion, when the temperatures of the peg and the block eventually equalize, the peg will expand and fill the hole tightly. For Scenario 2: Cooling the peg causes it to contract, allowing it to be inserted into the hole in the block, which is at a higher temperature. After insertion, when the temperatures of the peg and the block equalize, the peg will expand, and the hole in the block will contract; thus, they may not maintain the tight fit. After comparing both scenarios, it is clear that the best option is to heat the metal block (making the hole expand) while keeping the metal peg at a lower temperature during insertion. This will ensure a tight fit when the temperatures of the peg and the block eventually equalize.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Metal Peg and Hole Fitting
Fitting a metal peg into a hole snugly doesn't only require precision in size, but also an understanding of thermal expansion. When dealing with metal objects, it is crucial to recognize the relationship between temperature and the dimensions of metal. By manufacturing a peg slightly larger than the hole, we intentionally use temperature changes to manipulate sizes for a perfect fit. The goal is to adjust the temperature so that inserting the peg into the hole is possible without damaging either component. By changing the temperature of the block or the peg, we can alter their sizes temporarily to achieve this. In a practical scenario, two approaches are generally considered:
  • Heating the block to expand the hole.
  • Cooling the peg to contract it.
Both of these involve careful use of temperature control to ensure a seamless fit. Once the peg is inserted, the aim is to have the temperatures of the peg and the block reach equilibrium, so they both return to their original sizes while maintaining the tight fit.
Temperature Difference Effects
Temperature changes significantly affect materials, especially metals. This effect can be pivotal for fitting a metal peg into a metal block. By adjusting the temperatures of the peg and block, their dimensions can be tailored temporarily for the insertion process.

Heating the Block

When we heat the block, the molecules within the metal gain energy and move apart, thereby expanding the material. This expansion increases the size of the hole, making it easier to insert the slightly larger peg. As the block cools back to room temperature, the materials contract, ensuring a snug fit.

Cooling the Peg

Conversely, cooling the peg has the opposite effect. The molecules lose energy and slow down, causing the peg to contract slightly and allowing it to fit into the tighter hole. Once the peg returns to the ambient temperature, it expands back to its manufactured size, assuming the shape of the hole within the block. However, this scenario may not provide as secure of a fit if the hole also contracts slightly.
Expansion and Contraction in Solids
Expansion and contraction are natural responses of solids to temperature changes, making them predictable and useful in various applications like fitting metal pieces together. The process of thermal expansion relies on the basic principle that as the temperature of a solid increases, its particles gain energy and vibrate more vigorously, causing the material to expand.
  • At higher temperatures, atoms move more and need more space, causing expansion.
  • At lower temperatures, atoms slow their movements, resulting in contraction.

Applications in Manufacturing

This principle is particularly useful in manufacturing and repairs where perfect fits are necessary. Manufacturing an object to be slightly larger or smaller helps utilize this predictable behavior of materials. Overall, understanding these principles is crucial for tasks that involve precise fitting, showcasing how temperature can be a tool rather than just a condition.

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