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mortonsalt

  • one year ago

A fine steel wire is attached to an aluminum cylinder in parallel (see picture I'll attach below). The structure is assembled at 10.0°C under negligible stress. The initial length of the system is 85.0 cm long. The system is heated to 120°C. Calculate the resulting stress in the wire, assuming that the rod expands freely. Justify the assumption that the rod freely expands.

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  1. mortonsalt
    • one year ago
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    |dw:1443935742152:dw|

  2. mortonsalt
    • one year ago
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    Is the strain on the Al rod as it expands equal to the stress on the steel wire?

  3. mortonsalt
    • one year ago
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    Or do I use the formula \[\frac{F}{A} = -Y \alpha \Delta T\]

  4. Michele_Laino
    • one year ago
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    I think that we can assume that rod can expand freely since Aluminum is more sensible to temperature than steel, here are the values of linear expansion coefficients: Al---> 23.6 Steel (Fe)---> 11.7 the units are \(10^{-7} K^{-1}\)

  5. Michele_Laino
    • one year ago
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    oops.. the units are: \(10^{-6} K^{-1}\)

  6. mortonsalt
    • one year ago
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    Firstly, thank you! :) Is there any way I can use that information to calculate the resulting stress on the wire?

  7. Michele_Laino
    • one year ago
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    I'm thinking...

  8. mortonsalt
    • one year ago
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    Alrighty! Thanks!

  9. Michele_Laino
    • one year ago
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    we can compute the unitary stress, namely force over cross section of wire, using this formula: \[\sigma = \alpha \cdot \Delta t \cdot E\] where \(\alpha\) is the linear expansion coefficient of steel, \(\Delta t\) is the change of temperature, and \(E\) is the Young modulus, namely the same as your formula above

  10. Michele_Laino
    • one year ago
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    of corse \(E\) is the Young modulus of steel

  11. Michele_Laino
    • one year ago
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    of course*

  12. mortonsalt
    • one year ago
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    Alright. Here are my thoughts: The rod will expand faster than the wire. Is there another force pulling at the wire from the expansion of the rod? If so, would it affect how much stress is applied on the rod?

  13. mortonsalt
    • one year ago
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    Sorry, how much stress is applied on the wire*

  14. Michele_Laino
    • one year ago
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    I think that we have to consider the weight of the wire, nevertheless it can be considered negligible in comparison with the unitary stress \(\sigma\)

  15. mortonsalt
    • one year ago
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    Would that be it then? I just have to plug in the values into the formula for unitary stress?

  16. Michele_Laino
    • one year ago
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    yes! I think so!

  17. mortonsalt
    • one year ago
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    Alrighty! Thanks for the input!

  18. Michele_Laino
    • one year ago
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    :)

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