# Thermal Expansion of a metal ring

In summary, a ring of aluminum with a hole in the middle will expand when heated. The area of the hole will expand by the same percent as the area of the aluminum, not the linear expansion of the ring.

## Homework Statement

A ring of aluminum has a hole in the middle. When the ring is heated:

a) the hole decreases in diameter
b) the aluminum expands outward and the hole remains the same size.
c) the area of the hole expands by the same percent as the area of the aluminum.
d) the area of the hole expands by a greater percent than the area of the aluminum.
e) the linear expansion causes the hole to expand in a slightly elliptical pattern.

## Homework Equations

ΔA=$\alpha$AΔT
$\alpha$=area thermal expansion coefficient
A=area
T=temperature

## The Attempt at a Solution

I know the hole gets larger. What I can't figure out is whether the hole expands by the same percent of the area of by a greater percent. I want to say that it will expand at a greater percent because the metal expands everywhere in one dimension but area is in two dimensions (if that makes any sense).
Like the area of the circle is ∏r2, but the area of the ring is ∏(R2-r2) with big R being the outer radius and little r being the inner radius. Because of this formula, wouldn't the inside expand by a greater amount than the outside.
I've been thinking about this problem so long I've kinda gotten burnt out at looking at it from any other angles...

If 2πR and 2πr are the outer circumference and the inner circumference of the ring before the ring is heated, what are the inner and outer circumferences after it is heated? What are the inner and outer radii after it is heated? What is the area of the hole after the ring is heated? What is the area of the ring after it is heated?

Are those real questions or hints to get me going in the right direction?
I gave all the information given, this is a conceptual question...

I don't know how to compare the final to the initial, that's why I am having a hard time finding whether the answer is C) or D). I think I could be approaching the problem from the entirely wrong direction. Also there is very little information on these types of problems in my textbook.

(By no means do I intend to offend, I am pretty tired...)

c) the area of the hole expands by the same percent as the area of the aluminum.
d) the area of the hole expands by a greater percent than the area of the aluminum.
I want to say that it will expand at a greater percent because the metal expands everywhere in one dimension but area is in two dimensions (if that makes any sense).
It does, but the options c and d don't compare area of hole with linear expansion of ring; they compare it with area expansion of ring.

Is it valid to imagine this drawing as being a painting on a square sheet of metal? Then address the question, how will this painted pattern change as the metal sheet warms?

Are those real questions or hints to get me going in the right direction?
I gave all the information given, this is a conceptual question...

I don't know how to compare the final to the initial, that's why I am having a hard time finding whether the answer is C) or D). I think I could be approaching the problem from the entirely wrong direction. Also there is very little information on these types of problems in my textbook.

(By no means do I intend to offend, I am pretty tired...)

They are real questions and hints to get you going in the right direction.

Let me get you started. If 2πR is the outer circumference before the ring is heated, then after the ring is heated, the outer circumference is $2\pi R (1+\beta \Delta T)$, where β is the linear coefficient of thermal expansion. The answers to the rest of my equations should be easy now.

Chet

## 1. What is thermal expansion?

Thermal expansion refers to the increase in size or volume of a substance when its temperature increases.

## 2. How does thermal expansion affect metal rings?

When a metal ring is heated, its molecules vibrate more vigorously and take up more space, causing the ring to expand in size. Similarly, when the temperature decreases, the molecules move less and the ring contracts in size.

## 3. What factors affect the amount of thermal expansion in a metal ring?

The amount of thermal expansion in a metal ring depends on the type of metal, its initial size and temperature, and the change in temperature. Different metals have different coefficients of thermal expansion, which determine how much they will expand or contract.

## 4. Can thermal expansion cause damage to a metal ring?

Yes, thermal expansion can cause damage to a metal ring if the change in temperature is too extreme. This can lead to warping, cracking, or even breaking of the ring. It is important to consider the coefficient of thermal expansion when designing and using metal rings in applications where temperature changes are expected.

## 5. How is thermal expansion of a metal ring measured?

The thermal expansion of a metal ring can be measured using a device called a dilatometer, which measures the change in length of the ring as it is heated or cooled. The coefficient of thermal expansion can then be calculated using the change in length and the initial temperature of the ring.

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