(a) The scalar moment of inertia for a system of $N$ particles is given by

$$I=\sum_{i=1}^{N} m_{i} \mathbf{r}_{i} \cdot \mathbf{r}_{i}$$

where $m_{i}$ is the particle's mass and $\mathbf{r}_{i}$ is a vector giving the particle's position. Show that, for non-relativistic particles,

$$\frac{1}{2} \frac{d^{2} I}{d t^{2}}=2 K+\sum_{i=1}^{N} \mathbf{F}_{i} \cdot \mathbf{r}_{i}$$

where $K$ is the total kinetic energy of the system and $\mathbf{F}_{i}$ is the total force on particle

Assume that any two particles $i$ and $j$ interact gravitationally with potential energy

$$V_{i j}=-\frac{G m_{i} m_{j}}{\left|\mathbf{r}_{i}-\mathbf{r}_{j}\right|}$$

Show that

$$\sum_{i=1}^{N} \mathbf{F}_{i} \cdot \mathbf{r}_{i}=V$$

where $V$ is the total potential energy of the system. Use the above to prove the virial theorem.

(b) Consider an approximately spherical overdensity of stationary non-interacting massive particles with initial constant density $\rho_{i}$ and initial radius $R_{i}$. Assuming the system evolves until it reaches a stable virial equilibrium, what will the final $\rho$ and $R$ be in terms of their initial values? Would this virial solution be stable if our particles were baryonic rather than non-interacting? Explain your answer.