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.

Question

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.

mortonsalt · Answer

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mortonsalt · Answer

Is the strain on the Al rod as it expands equal to the stress on the steel wire?

mortonsalt · Answer

Or do I use the formula $$\frac{F}{A} = -Y \alpha \Delta T$$

michele_laino · Answer

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 $10^{-7}  K^{-1}$

michele_laino · Answer

oops..
the units are: $10^{-6} K^{-1}$

mortonsalt · Answer

Firstly, thank you! :) 

Is there any way I can use that information to calculate the resulting stress on the wire?

michele_laino · Answer

I'm thinking...

mortonsalt · Answer

Alrighty! Thanks!

michele_laino · Answer

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

michele_laino · Answer

of corse $E$ is the Young modulus of steel

michele_laino · Answer

of course*

mortonsalt · Answer

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?

mortonsalt · Answer

Sorry, how much stress is applied on the wire*

michele_laino · Answer

I think that we have to consider the weight of the wire, nevertheless it can be considered negligible in comparison with the unitary stress $\sigma$

mortonsalt · Answer

Would that be it then? I just have to plug in the values into the formula for unitary stress?

michele_laino · Answer

yes! I think so!

mortonsalt · Answer

Alrighty! Thanks for the input!

michele_laino · Answer

:)