# Thermodynamics Linear Expansion Problem

• Ling_Ling
In summary, a brass plug and iron ring with different coefficients of linear expansion need to be brought to a common temperature in order to fit. By setting the initial and final lengths of both objects equal to each other, the temperature can be solved for using the equation L = Lo*(1+α⌂T). After attempting multiple methods, it was determined that the final temperature should be 183.05°C.
Ling_Ling

## Homework Statement

A brass plug is to be placed in a ring made of iron. At room temperature (20°C), the diameter of the plug is 8.749 cm and that of the inside of the ring is 8.739 cm. They must both be brought to what common temperature in order to fit?

It's a linear expansion problem. As a a ring is heated, the diameter of the hole expands.
d-plug = 8.749 cm
d-ring = 8.739 cm
⌂diameter = .01cm
α(brass) = 19E-6
α(iron) = 12E-6
⌂T = 20-T or 293-T (⌂T is constant)
To = 20°C = 293°K (Which one should I use?)

T = ?

L = Lo*(1+α⌂T)

## The Attempt at a Solution

Set the L's of the brass and iron rings equal to each other.
Lo*(1+α⌂T)=Lo*(1+α⌂T)
8.749(1+19E-6*⌂T) = 8.739(1+12E-6*⌂T)
8.749 + .000166⌂T = 8.739 + .000105⌂T
.01 = -.000061⌂T
⌂T = -163.05 °C (I think it's Celsius)
20-(-163.05) = 183.05°C

I attempted a few different methods to solving this, but they proved to be wrong. Am I correct in my thinking?

You work looks good to me.

Okay, thank you. I just needed to check that I had the right work and thinking.

## 1. How does thermal expansion affect different materials?

Thermal expansion is the tendency of materials to expand or contract when exposed to heat. Different materials have different coefficients of thermal expansion, meaning they will expand or contract at different rates.

## 2. What is the formula for linear thermal expansion?

The formula for linear thermal expansion is ΔL = αL₀ΔT, 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.

## 3. How does temperature affect the length of an object?

As the temperature of an object increases, its molecules vibrate and move more rapidly, causing it to expand in length. As the temperature decreases, the molecules slow down and the object will contract in length.

## 4. What is the coefficient of linear expansion?

The coefficient of linear expansion is a measure of how much a material will expand or contract in length for every degree change in temperature. It is typically measured in units of length per degree Celsius (or Kelvin).

## 5. How is thermal expansion used in everyday life?

Thermal expansion is used in everyday life in a variety of ways, such as in the construction of bridges and buildings. It is also used in thermometers, thermostats, and other devices that measure temperature. Many household objects, such as metal pots and pans, also expand and contract with changes in temperature.

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