## Question:

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 a higher temperature?

## Step-by-step

As a result of thermal expansion, the size of both spheres increases. The center of mass of the sphere lying on the plate rises, whilst that of the sphere hanging on the thread sinks. Thus, the potential energy of the first sphere increases, whilst that of the second one decreases as shown in the figure.
According to the first law of thermodynamics, the heat transferred to the spheres produces not only an increase in internal energy and a small amount of work done in expanding against the atmospheric pressure (this is the same for both spheres) but also a change in gravitational potential energy. The potential energy of the sphere lying on the insulating plate increases a little, therefore its internal energy increases by less than the residual heat transferred. Conversely, the decrease in potential energy of the hanging sphere contributes positively to the increase in its internal energy. In summary, the temperature of the sphere suspended from the thread will be higher. It is worth giving a numerical estimate. If the temperature of the two iron balls, each with a radius of 10 cm, is increased by ${100}^{\omicron }$ C, a temperature difference of $∆T ≈ 5×{{10}^{-6{\omicron }}}$ C will result from this effect. This is undetectable in practice.