# Iron Ring Expansion: Solve for Δθ to Find Temp Increase

• baldbrain
In summary: Thanks for pointing it out. In summary, the problem involves fitting an iron ring of radius 2.1 m onto the rim of a wheel of radius 2.121 m, and the coefficient of volume expansion for iron is given as 3.6 x 10-5 K-1. To find the temperature increase needed for the ring to fit, the linear thermal expansion coefficient must be used instead, which is related to the volume expansion coefficient by the equation γ=3α. The change in temperature can be calculated in either units of K or °C.
baldbrain

## Homework Statement

An iron ring of radius 2.1 m is to be fitted on the rim of a wheel of radius 2.121 m. The coefficient of volume expansion for iron is 3.6 x 10-5 K-1. By how much should the temperature of the ring be increased?
(a) 532 °C
(b) 833 °C
(c) 278 °C
(d) 378 °C

## Homework Equations

ΔV/V = γΔθ
where ΔV in the change in volume on expansion, V is the original volume, γ is the coefficient of volume expansion & Δθ is the rise in temperature.
3. The attempt at a solution
Well, obviously, ΔV = 0.021 m. Now, if I use the value of γiron in (°C)-1, that is, 12 x 10-6 (not given in the problem), I get the correct answer of 833 °C. However, if I use the given value of 3.6 x 10-5 K-1, I get the answer as 277.78 K, which comes out to be approximately 4 °C.
Why is there a discrepancy?
How do I convert γiron in K-1 to (°C)-1?

baldbrain said:

## Homework Statement

An iron ring of radius 2.1 m is to be fitted on the rim of a wheel of radius 2.121 m. The coefficient of volume expansion for iron is 3.6 x 10-5 K-1. By how much should the temperature of the ring be increased?
(a) 532 °C
(b) 833 °C
(c) 278 °C
(d) 378 °C

## Homework Equations

ΔV/V = γΔθ
where ΔV in the change in volume on expansion, V is the original volume, γ is the coefficient of volume expansion & Δθ is the rise in temperature.
3. The attempt at a solution
Well, obviously, ΔV ΔL= 0.021 m. Now, if I use the value of γiron in (°C)-1, that is, 12 x 10-6 (not given in the problem), I get the correct answer of 833 °C. However, if I use the given value of 3.6 x 10-5 K-1, I get the answer as 277.78 K, which comes out to be approximately 4 °C.
Why is there a discrepancy?
How do I convert γiron in K-1 to (°C)-1?
The problem gives the volume expansion coefficient. You have to fit the ring onto the rim of the wheel, so the length should be increased. The volume of the ring is irrelevant.
You have to calculate the relative change of length, and you have to use the linear thermal expansion coefficient. How is it related to the volume expansion coefficient?
In the formula for the thermal expansion, you have the change of temperature. The temperature difference is the same both in K and °C.

Last edited:
ehild said:
The problem gives the volume expansion coefficient. You have to fit the ring onto the rim of the wheel, so the length should be increased. The volume of the ring is irrelevant.
You have to calculate the relative change of length, and you have to use the linear thermal expansion coefficient. How is it related to the volume expansion coefficient?
Oh, holy cow! Change in circumference, you're damn right! I'm a dumbass.
γ=3α. I get it now.
ehild said:
In the formula for the thermal expansion, you have the change of temperature. The temperature difference is the same both in K and °C.
I missed that too

## 1. What is the Iron Ring Expansion?

The Iron Ring Expansion is a scientific concept that describes the change in temperature of a metal object when it is heated and expands.

## 2. How is Δθ used in the Iron Ring Expansion equation?

Δθ, or delta theta, represents the change in temperature in degrees Celsius. It is used in the equation to calculate the temperature increase of the metal object.

## 3. What is the formula for Iron Ring Expansion?

The formula for Iron Ring Expansion is Δθ = LαΔT, where Δθ is the change in temperature, L is the length of the metal object, α is the coefficient of linear expansion, and ΔT is the change in temperature in degrees Celsius.

## 4. How do you solve for Δθ in the Iron Ring Expansion equation?

To solve for Δθ, you need to know the values of L, α, and ΔT. Plug these values into the equation and solve for Δθ using basic algebraic principles.

## 5. What is the significance of Iron Ring Expansion in scientific research?

Iron Ring Expansion is an important concept in materials science and engineering. It helps scientists and researchers understand how different materials react to changes in temperature and how they can be used in various applications.

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