Equal heat added to two spheres: why is the hanging sphere hotter?

[sol_2001]

Homework Statement

Consider two identical iron spheres, one of which lies on a thermally insulating plate, whilst the other hangs from an insulating thread. Equal amounts of heat are given to the two spheres. Which will have the higher temperature?

Relevant Equations

I am not sure which equations are most relevant yet. My guess is that this involves thermal expansion, conservation of energy, and possibly gravitational potential energy, but I do not yet know how to connect them properly.

My initial thought was that the hanging sphere would end up at a higher temperature because the contact area between the sphere and the insulating thread is much smaller than the contact area between the other sphere and the insulating plate. I thought that if heat could be exchanged through the supports, the sphere on the plate might transfer energy away more easily because of the larger contact area.
However, I am confused because the problem explicitly says the plate and thread are thermally insulating, so I assume heat transfer through the supports is meant to be ignored.
What I am struggling to understand is:

• why the arrangement matters if equal amounts of heat are added to both identical spheres
• why contact area is not the deciding factor here
• how thermal expansion, centre of mass, and gravitational potential energy are relevant to the temperature difference

I am looking for a conceptual explanation more than just the final answer, since I have not yet been taught how to connect those ideas in thermodynamics.

 

[.Scott]

It's hard to know what the homework problem is trying to get at. But I would note that even a hot sphere floating in a vacuum will loose heat - more "wrapping" would reduce that. On the other hand, since more of the insulating material is in proximity to the heated sphere, more of it will be heated. So, an argument can be made both ways.
The way that the question is posed, there is no delay from the time that the "heat is given" to the time that the temperatures are compared. No time for radiative cooling, no time for heat conduction into the insulating material. But it does mention a particular material (iron), so I would go with the thermal expansion analysis.

 

[kuruman]

This problem is on many websites and you can find an answer and explanation without looking too hard or having to subscribe or pay money to someone. I don't know if you have seen the solution and are asking for further clarifications or you are looking a starting point. In case it is the latter and having seen the answer, I recommend that you start with the following assumptions:

• The spheres are isolated and do not exchange heat with one another.
• No heat is lost by conduction to the supporting plate or suspension thread.
• The spheres are in vacuum so there is no convection to consider.
• A sphere does not lose heat by radiation to the chamber that contains it.

Given all that, consider that total energy is conserved, so each sphere receives the same $Q$ Joules, where do these Joules go in each case?

As @.Scott said,

. . . go with the thermal expansion analysis.