Expansion of ruler and rod help

[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! :)

 

[asteriatic]

So I tried using that and the initial length of the rod to find the change in length for the ruler (using the same equation), but I'm not sure if that is the correct approach.

I would approach this problem by first understanding the concept of thermal expansion. I would know that when an object is heated, it expands due to the increased kinetic energy of its molecules. This expansion can be quantified by the coefficient of expansion, which is a characteristic property of the material.

To solve part a, I would use the equation given in the problem, but I would also take into account the expansion of the ruler. Since both the rod and ruler are made of steel, they would have the same coefficient of expansion. So, I would set up an equation with two unknowns - the coefficient of expansion and the change in length of the ruler. Then, I would use the given values and solve for the coefficient of expansion.

Once I have the coefficient of expansion for steel, I can use it to solve parts b and c. For part b, I would use the same equation but with the new length of the rod (twice the original length). This would give me the new coefficient of expansion for the longer rod.

For part c, I would use the same approach as part b, but I would also need to consider the change in diameter. The coefficient of expansion for a material is dependent on its shape and dimensions, so a rod with twice the diameter would have a different coefficient of expansion than a rod with the same length but half the diameter.

In conclusion, understanding the concept of thermal expansion and using the appropriate equations and values would help in solving this problem. It is important to also consider the properties of the material and how they may affect the coefficient of expansion in different scenarios.