Thermal expansion geometry Suppose I have a metal coin on a table. Let's make it spicy by assuming the coin is a perfect cylinder, fully metallic, but with non-uniform mass distribution, such that its centre of mass is not on the cylinder's axis. If I raise the room's temperature, it will expand thermally. With only the usual forces acting on it (gravity,normal), what will its final position be on the table?

*

*Will the CM stay still?

*Will the cylinder's axis stay still?

*Something else?

PS tag is not very good, couldn't find "thermal-expansion", is there something similar?
 A: Since the coin is made of multiple metals which are not alloyed, as the centre of mass is not on the cylindrical axis, the relative position of the centre of mass would change. This would happen when the metals have different coefficients of thermal expansion. One would expand more than the other and would distort the geometry of the coin. So the COM would move.
Another example, where the COM undergoes a change in relative position is a bimetallic strip. Upon expansion, the bimetallic strip bends towards the side of the metal having a smaller coefficient of thermal expansion ($\alpha$), and hence a change in the position of the COM occurs.
A: To simplify the analysis, consider a one-dimensional component with an initial length $L_o$ (m) along a $z$ direction starting at a temperature $T_o$.
A distribution of mass in a component arises from a distribution of linear mass density $\rho_L(z)$ (kg/m). This can happen in two ways. One is that a pure substance has a distribution of atomic or molecular separation distances. We can observe such behavior in a glassy polymer, where the distances between the molecular chains can vary, for example due to anisotropic stress gradients during formation. The second way we obtain a distribution of mass in a component is when the component is comprised of different substances that are non-uniform in composition as a function of position. Consider a metal rod of two components (i.e. brass) where the composition of the zinc varies along the length of the rod. This is not necessarily because the metals not alloyed throughout the rod (in direct counter to what is suggested in the other answer). This is simply a case where composition varies as a function of position, yet each position can still be a single, solution phase (a distinguishing characteristic of metals being called alloys).
Regardless of why, the center of mass $Z_{CoM}$ (m) of a linear rod of total mass $m_T$ is found from
$$ Z_{CoM} = \frac{\int_o^{L_f} \rho_L(z)\ z\ dz}{m_T} $$
How do we find the position as a function of temperature? The temperature dependence enters because the length of the rod changes through the thermal expansion coefficient $\alpha_L$ (1/K). The definition in a 1-D frame is
$$ \alpha_L \equiv \frac{1}{l}\frac{dl}{dT} $$
Consider the case where $\alpha_L$ is constant even though $\rho_L$ varies with $z$. As temperature changes, we can determine how $z(T)$ varies in relation to the reference temperature by
$$ z = z_o \exp\left( \alpha_L \Delta T \right)$$
Substitute back to the founding equation to obtain the result
$$ Z_{CoM,\star} = \frac{z_o\ \exp\left( \alpha_L \Delta T \right)}{m_T} \int_{L_o}^{L_f}\ \rho_L(z)\ dz $$
The CoM will change its $z$ position along the rod depending on the density distribution.
Analysis is more difficult when thermal expansion coefficient depends on position $\alpha_L(z)$, as must be taken to be typical for the real case where density depends on position $\rho_L(z)$. The development in such cases is left as an exercise, perhaps for a follow up Stack Exchange question. Finally, the full analysis of the coin must be done in a cylindrical coordinate system with $\rho_L(r,\theta)$. This also adds complexity to the mathematics. The insights from the linear rod remain entirely valid.
