Finding coefficient of expansion when ruler also expands

[rdn98]

The length of a metal rod is measured to be 20.08 cm using a steel ruler when both the rod and the ruler are at 22oC. Both the rod and the ruler are raised to a temperature of 253oC. When the rod is measured at this higher temperature, its length is found to be 20.32 cm.

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a) What is the coefficient of expansion of the metal?
b) You now make a rod of the same material but with twice the length. What will it coefficient of expansion be?
c) You now make a rod of the same material but with twice the diameter. What will it coefficient of expansion be?
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So far I am stuck on part a. If I can get this part, then I should be able to get the rest.

Now I know that the equation is delta L=L*(coefficient of expansion)*(delta T)

Now it appears that I have all the informatin given in the problem, so I plug it into the equation, but it doesn't work. Then I realize that the ruler is stretching also, which means the number scale on the ruler also changes. Now I have to somehow relate that number scale to the coefficient of expansion, but I'm not sure how.

I know from the book at the coefficient of expansion for steel is 11*10^-6 /degree C.

 

[Doc Al]

Originally posted by rdn98
Then I realize that the ruler is stretching also, which means the number scale on the ruler also changes. Now I have to somehow relate that number scale to the coefficient of expansion, but I'm not sure how.

First find the actual length of the expanded rod. You know the length of the expanded rod as measured by the expanded steel ruler. So, knowing the expansion of the steel, find the actual length of each "1 cm" marking on the ruler. Then you can convert to real units.

Hint: Write down the final expression (for the coefficient of expansion of the rod) and simplify it before you start doing any arithmetic.

 

[Rapier]

Thread necromancy!

I've got slightly different numbers: The length of a metal rod is measured to be 20.08 cm using a steel (α = 1.1e-005 (oC)-1) ruler when both the rod and the ruler are at 22oC. Both the rod and the ruler are raised to a temperature of 253oC. When the rod is measured at this higher temperature, its length is found to be 20.32 cm.

When heated, the rod expands. However, the ruler also expands. So the rod doesn't actually expand from 20.08 to 20.32. I know the rod expands more than the ruler, because after heating, the rod is measured to be longer. If they expanded the same the length wouldn't change and if the rod expanded less the final length would be less than the original length.


I used 1 cm because I need to find the actual length of each '1 cm marking' after heating.
ΔLs = L $\alpha$ Δt
ΔLs = (1 cm) (1.1e-5 °C^-1) (253°C - 20°C)
ΔLs = .002563
1 hcm (heated cm) = 1.002563 cm (real cm)

If the rod now measures 20.32 hcm, so I convert my units:

20.32 hcm * (1.002563 cm/1 hcm) = 20.37208 cm

So now I have an actual length and can calculate the $\alpha$.

$\alpha$ = ΔL/(L*Δt)
$\alpha$ = .00435

And it tells me NO!

 

[Rapier]

Doh!

If I actually use a ΔL instead of L it works out.

/alpha = 6.243e-5

Thanks for being here so I can think this through! :)