Thermodynamics and Mechanics: Complementarity

Sometimes separate fields in Physics come together for an unified application in certain problems. Don't believe me? Just take a look at the following thought experiment:

We have two identical metal spheres (consider them perfectly spherical!) with radius r, mass m, in the presence of a gravitational field on Earth g - but one of them hangs on a thread of negligible thickness and size, while the other lies motionless on the ground (assume that the ground is rigid and doesn't absorb any heat from anything including the ball). Now, the ball has a heat capacity c, which we can assume to be constant throughout this experiment (of course it won't be, but why torture ourselves here?)

The experimental procedure is as follows: we use a flame to impart E joules of energy to both balls and simply wait.

The question I have for you is: What is the final temperature of both balls? Will they be the same? If yes why? If not, why? [Hint: Of course the final temperature won't be the same!]

Here's another 2 questions for the really psychedelic hardcore Physics lover: Can you work out for me the difference in temperature? Can you also work out the difference in size? [Assume that the ball has a constant expansivity of a]

Working this problem out just demonstrates how intricately thermodynamics is linked to mechanics; indeed, the First Law of Thermodynamics itself embodies the essence of one of the important principles in mechanics, the Principle of Conservation of Energy. :)

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Well, it's been days, and no one has replied or commented or whatsoever, so I guess it's time to go on and reveal more of the answer to fulfill my own needs and complete the post. This question is a very simple one, as long as one bothers to draw out the initial state and final state of the system - notice that I'm making use of one of the basic concepts in Thermodynamics, the fact that initial and final states determine the change in any state function, and in this case, the temperature of the ball.

Consider the two expansion processes:

What do you notice? I'll leave this post hanging here for another few days before someone comes along to comment on what happens. :p