Rock On Rod Stewart: Am I a Universe-Rule-Changer?

[one_q_a_day]

Experiment 1:
I have a steel rod . I heat the rod to temperature T. As the rod heats up, it expands to length L. This makes me happy.

I let the rod cool down and contract back to its original temperature and dimensions.

Experiment 2:
Now I place a heavy rock on top of the rod, and heat the rod again to temperature T.

http://en.wikipedia.org/wiki/Thermal_expansion#Expansion_in_solids". so (trusting wikipedia) i know that the rod expands to length L again.

the rod's expansion lift the rock slightly.

Q: Do I need the same amount of energy for E1 and E2?
if the answer to Q is No, then i tricked the rules of the universe, and raised the potential energy of the rock with zero energy

if the answer to Q is Yes, then i tricked the rules of the universe, and changed the specific heat of the rod just by placing a rock on top of it.

Am i a universe-rule-changer or is my logic faulty?

 

[James Leighe]

The bar without the rock on top of it will be longer than the one with a rock on it.

But only by a super tiny amount.

 

[D H]

Am i a universe-rule-changer or is my logic faulty?

Without math you are not doing physics. Why don't you try doing the math instead of using faulty logic?

 

[sophiecentaur]

It's far too late in the day to try to find holes this sort of basic Physics. (Fun trying, tho' and fun answering).
Work done will equal energy supplied.
Work is force times distance.
Raising the rock so that the rod will end up the same length will require the rod to be at a higher temperature because the atoms will need to vibrate (in very crude terms) more in order to overcome the compressive force in addition to fighting their mutual attractive forces (which is why you get expansion in the first place) - that alone will tell you that you need more energy.
Next...

 

[nasu]

Experiment 1:
i know that the rod expands to length L again.

How do you know this? You can actually "force" the rod to keep its original length at the higher temperature, if you apply sufficient external force. So this is not obvious and probably not true.
Besides, when you add the rock on top, the rod is compressed so the initial (before heating) length decreases. You are not in the same initial condition as without the rock.
Anyway, when you have a constraining external force applied, you'll need a higher temperature to get the same final length.

 

[sophiecentaur]

It's all down to work = force times distance

put in enough energy and the appropriate work will be done - rod can reach same length.

The devil is in the detail; Wikkers says expansion is BARELY affected - not that it ISN'T affected by the pressure. For the same length, you need a higher temperature.
This is true even if the rock isn't actually resting on the rock at the start.

 

[one_q_a_day]

Without math you are not doing physics. Why don't you try doing the math instead of using faulty logic?

You're right.

Lets say the rod (height: 1m; top area: 100cm²) is made of steel (thermal expansion coefficient: 0.00001/C at 20C; specific heat capacity: 0.5 kJ/kgC; density: 8g/cm³)

The rock's mass is 1000kg

rod volume = 1m * 100cm² = 10000cm³
rod mass = 10000cm * 8g/cm³ = 80kg

now let's design 3 new experiments (slightly different from original post):

E3 (rod): place rod on Earth at room temperature (20C, 1atm), heat it so that its height is increased by 1mm

E4 (rock): place rock on earth, lift it by 1mm

E5 (rock on rod): place rod on earth, rock on rod. heat up rod so that its height is increased by 1mm

now let's see how much work we need for each experiment

E3:
lets see what temperature increase is needed in order to achieve a height increase of 1mm:
(1mm/1m) / (0.00001/C) = 100C

how much work is needed in order to raise the rod's temperature by 100C:
80kg * 100C * 0.5kJ/kgC = 4000kJ

E4:
how much work is needed to lift the 1000kg rock by 1mm:
1000kg * 9.8 m/s² * 0.001m = 9.8J

E5:
what's the pressure on the rod exerted by the rock:
1000kg * 9.8m/s² / 100cm² = 980kPa ~ 9.8atm

lets figure out the work needed to increase the height of the rod by 1 mm when the 1000kg is on top is (given the principle of the 'conservation of energy'):
4000kJ + 9.8J = 4000009.8J

indeed, the 9.8atm pressure exerted by rock on the rod requires just a little bit more work to achieve the 1mm rod height increase

now let's perform another experiment. let's hollow out the rod by removing 99% of the material. were are left with a cylinder whose mass is 0.8kg and top surface area is 1cm.

E6 (rock on cylinder): place cylinder on earth, rock on cylinder. heat up cylinder so that its height is increased by 1mm

how much work is to be done for E6?
40kJ + 9.8J = 40009.8J

the pressure exerted by the rock on the cylinder is now about 980atm. i believe the steel cylinder can withstand this.

now the extra pressure requires us to work about 0.025% harder. is this an appreciable amount?

Questions:
1. do you see any error in my calculations?
2. can you point me to sources describing how thermal expansion coefficient of solids is affected by pressure?

 

[sophiecentaur]

There is another aspect to this problem.
The rod expands in all three dimensions. If there is extra stress in one direction they the situation is no longer symmetrical. Some energy will also be stored in the rod as it expands more laterally than it would if it were unloaded. A very small distance but a total volume which is of the same order as the extra volume on the end of the rod and an increase in internal stress / stored energy.

In a similar scenario involving a gas balloon with a mass on top of it, we'd automatically talk in terms of volume expansion and not linear expansion. The use of linear expansion could just be inadequate in this case. I can't think how to work out the answer for a solid because it must involve some matrix (/tensor?) cleverness but I think I've identified a direction to go in answering question 2 in the above post.

 

[Phrak]

Instead of all the speculating, look at the Poison ratio that tells you how pressure from the rock on the rod effects its volume. In this one dimensional case it gets a little tricky, so you also need the elastic modulus.

A much simpler thought experiment would be to heat a cube of material bound on all six sides by plates held against the heated sample by springs. We pretend that the plates don't change temperature or conduct heat. Now ask where the potential energy in the springs comes from when the sample expands under heating.

The control sample is the same cube without the springs and plates.

 

[sophiecentaur]

As I fell to the floor choking and clutching my throat, I realized you meant Poisson